# Exact Coordinate Instances in Fixed Real Vector Spaces

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261022-r04-real-vector-space-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-12, -3, 2], right=[-2, -8, 0]. Give (left,right,the componentwise sum).

View answer and explanation([-12, -3, 2], [-2, -8, 0], [-14, -11, 2])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, -3, 2], [-2, -8, 0], [-14, -11, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0001/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 2. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-11, -2, 1], right=[-1, -7, 1]. Give (left,right,the componentwise sum).

View answer and explanation([-11, -2, 1], [-1, -7, 1], [-12, -9, 2])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, -2, 1], [-1, -7, 1], [-12, -9, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0002/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 3. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-10, -1, 0], right=[0, -6, 2]. Give (left,right,the componentwise sum).

View answer and explanation([-10, -1, 0], [0, -6, 2], [-10, -7, 2])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, -1, 0], [0, -6, 2], [-10, -7, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0003/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 4. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-9, 0, -1], right=[1, -5, 0]. Give (left,right,the componentwise sum).

View answer and explanation([-9, 0, -1], [1, -5, 0], [-8, -5, -1])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 0, -1], [1, -5, 0], [-8, -5, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0004/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 5. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-8, 1, -2], right=[2, -4, 1]. Give (left,right,the componentwise sum).

View answer and explanation([-8, 1, -2], [2, -4, 1], [-6, -3, -1])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 1, -2], [2, -4, 1], [-6, -3, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0005/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 6. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-7, 2, -3], right=[-2, -3, 2]. Give (left,right,the componentwise sum).

View answer and explanation([-7, 2, -3], [-2, -3, 2], [-9, -1, -1])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, 2, -3], [-2, -3, 2], [-9, -1, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0006/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 7. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-6, 3, -4], right=[-1, -2, 0]. Give (left,right,the componentwise sum).

View answer and explanation([-6, 3, -4], [-1, -2, 0], [-7, 1, -4])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, 3, -4], [-1, -2, 0], [-7, 1, -4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0007/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 8. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-5, -3, -5], right=[0, -1, 1]. Give (left,right,the componentwise sum).

View answer and explanation([-5, -3, -5], [0, -1, 1], [-5, -4, -4])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, -3, -5], [0, -1, 1], [-5, -4, -4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0008/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 9. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-4, -2, -6], right=[1, 0, 2]. Give (left,right,the componentwise sum).

View answer and explanation([-4, -2, -6], [1, 0, 2], [-3, -2, -4])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, -2, -6], [1, 0, 2], [-3, -2, -4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0009/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 10. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-3, -1, -7], right=[2, 1, 0]. Give (left,right,the componentwise sum).

View answer and explanation([-3, -1, -7], [2, 1, 0], [-1, 0, -7])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, -1, -7], [2, 1, 0], [-1, 0, -7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0010/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 11. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-2, 0, -8], right=[-2, 2, 1]. Give (left,right,the componentwise sum).

View answer and explanation([-2, 0, -8], [-2, 2, 1], [-4, 2, -7])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, 0, -8], [-2, 2, 1], [-4, 2, -7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0011/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 12. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[-1, 1, -9], right=[-1, 3, 2]. Give (left,right,the componentwise sum).

View answer and explanation([-1, 1, -9], [-1, 3, 2], [-2, 4, -7])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, 1, -9], [-1, 3, 2], [-2, 4, -7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0012/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 13. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[0, 2, -10], right=[0, 4, 0]. Give (left,right,the componentwise sum).

View answer and explanation([0, 2, -10], [0, 4, 0], [0, 6, -10])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 2, -10], [0, 4, 0], [0, 6, -10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0013/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 14. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[1, 3, -11], right=[1, 5, 1]. Give (left,right,the componentwise sum).

View answer and explanation([1, 3, -11], [1, 5, 1], [2, 8, -10])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 3, -11], [1, 5, 1], [2, 8, -10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0014/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 15. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[2, -3, -12], right=[2, 6, 2]. Give (left,right,the componentwise sum).

View answer and explanation([2, -3, -12], [2, 6, 2], [4, 3, -10])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, -3, -12], [2, 6, 2], [4, 3, -10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0015/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 16. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[3, -2, -13], right=[-2, 7, 0]. Give (left,right,the componentwise sum).

View answer and explanation([3, -2, -13], [-2, 7, 0], [1, 5, -13])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, -2, -13], [-2, 7, 0], [1, 5, -13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0016/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 17. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[4, -1, -14], right=[-1, 8, 1]. Give (left,right,the componentwise sum).

View answer and explanation([4, -1, -14], [-1, 8, 1], [3, 7, -13])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, -1, -14], [-1, 8, 1], [3, 7, -13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0017/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 18. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[5, 0, -15], right=[0, 9, 2]. Give (left,right,the componentwise sum).

View answer and explanation([5, 0, -15], [0, 9, 2], [5, 9, -13])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0, -15], [0, 9, 2], [5, 9, -13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0018/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 19. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[6, 1, -16], right=[1, 10, 0]. Give (left,right,the componentwise sum).

View answer and explanation([6, 1, -16], [1, 10, 0], [7, 11, -16])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 1, -16], [1, 10, 0], [7, 11, -16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0019/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 20. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[7, 2, -17], right=[2, 11, 1]. Give (left,right,the componentwise sum).

View answer and explanation([7, 2, -17], [2, 11, 1], [9, 13, -16])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 2, -17], [2, 11, 1], [9, 13, -16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0020/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 21. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[8, 3, -18], right=[-2, 12, 2]. Give (left,right,the componentwise sum).

View answer and explanation([8, 3, -18], [-2, 12, 2], [6, 15, -16])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 3, -18], [-2, 12, 2], [6, 15, -16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0021/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 22. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[9, -3, -19], right=[-1, 13, 0]. Give (left,right,the componentwise sum).

View answer and explanation([9, -3, -19], [-1, 13, 0], [8, 10, -19])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, -3, -19], [-1, 13, 0], [8, 10, -19]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0022/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 23. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[10, -2, -20], right=[0, 14, 1]. Give (left,right,the componentwise sum).

View answer and explanation([10, -2, -20], [0, 14, 1], [10, 12, -19])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, -2, -20], [0, 14, 1], [10, 12, -19]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0023/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 24. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[11, -1, -21], right=[1, 15, 2]. Give (left,right,the componentwise sum).

View answer and explanation([11, -1, -21], [1, 15, 2], [12, 14, -19])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, -1, -21], [1, 15, 2], [12, 14, -19]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0024/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 25. Coordinate Addition in a Fixed Real Space

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. left=[12, 0, -22], right=[2, 16, 0]. Give (left,right,the componentwise sum).

View answer and explanation([12, 0, -22], [2, 16, 0], [14, 16, -22])

Require matching dimensions, then add corresponding coordinates without changing their order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 0, -22], [2, 16, 0], [14, 16, -22]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0025/)

Real vector space · qf-str22-vectorspace-addition · candidate





## 26. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[-12, -2, 3]. Give (scalar,vector,scaled vector).

View answer and explanation(-3, [-12, -2, 3], [36, 6, -9])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [-12, -2, 3], [36, 6, -9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0026/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 27. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[-11, -1, 2]. Give (scalar,vector,scaled vector).

View answer and explanation(-2, [-11, -1, 2], [22, 2, -4])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [-11, -1, 2], [22, 2, -4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0027/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 28. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[-10, 0, 1]. Give (scalar,vector,scaled vector).

View answer and explanation(-1, [-10, 0, 1], [10, 0, -1])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [-10, 0, 1], [10, 0, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0028/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 29. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[-9, 1, 0]. Give (scalar,vector,scaled vector).

View answer and explanation(0, [-9, 1, 0], [0, 0, 0])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [-9, 1, 0], [0, 0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0029/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 30. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[-8, 2, -1]. Give (scalar,vector,scaled vector).

View answer and explanation(1, [-8, 2, -1], [-8, 2, -1])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, [-8, 2, -1], [-8, 2, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0030/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 31. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=2, vector=[-7, -2, -2]. Give (scalar,vector,scaled vector).

View answer and explanation(2, [-7, -2, -2], [-14, -4, -4])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, [-7, -2, -2], [-14, -4, -4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0031/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 32. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[-6, -1, -3]. Give (scalar,vector,scaled vector).

View answer and explanation(3, [-6, -1, -3], [-18, -3, -9])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, [-6, -1, -3], [-18, -3, -9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0032/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 33. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[-5, 0, -4]. Give (scalar,vector,scaled vector).

View answer and explanation(-3, [-5, 0, -4], [15, 0, 12])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [-5, 0, -4], [15, 0, 12]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0033/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 34. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[-4, 1, -5]. Give (scalar,vector,scaled vector).

View answer and explanation(-2, [-4, 1, -5], [8, -2, 10])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [-4, 1, -5], [8, -2, 10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0034/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 35. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[-3, 2, -6]. Give (scalar,vector,scaled vector).

View answer and explanation(-1, [-3, 2, -6], [3, -2, 6])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [-3, 2, -6], [3, -2, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0035/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 36. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[-2, -2, -7]. Give (scalar,vector,scaled vector).

View answer and explanation(0, [-2, -2, -7], [0, 0, 0])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [-2, -2, -7], [0, 0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0036/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 37. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[-1, -1, -8]. Give (scalar,vector,scaled vector).

View answer and explanation(1, [-1, -1, -8], [-1, -1, -8])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, [-1, -1, -8], [-1, -1, -8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0037/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 38. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=2, vector=[0, 0, -9]. Give (scalar,vector,scaled vector).

View answer and explanation(2, [0, 0, -9], [0, 0, -18])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, [0, 0, -9], [0, 0, -18]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0038/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 39. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[1, 1, -10]. Give (scalar,vector,scaled vector).

View answer and explanation(3, [1, 1, -10], [3, 3, -30])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, [1, 1, -10], [3, 3, -30]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0039/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 40. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[2, 2, -11]. Give (scalar,vector,scaled vector).

View answer and explanation(-3, [2, 2, -11], [-6, -6, 33])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [2, 2, -11], [-6, -6, 33]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0040/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 41. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[3, -2, -12]. Give (scalar,vector,scaled vector).

View answer and explanation(-2, [3, -2, -12], [-6, 4, 24])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [3, -2, -12], [-6, 4, 24]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0041/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 42. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[4, -1, -13]. Give (scalar,vector,scaled vector).

View answer and explanation(-1, [4, -1, -13], [-4, 1, 13])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [4, -1, -13], [-4, 1, 13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0042/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 43. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[5, 0, -14]. Give (scalar,vector,scaled vector).

View answer and explanation(0, [5, 0, -14], [0, 0, 0])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [5, 0, -14], [0, 0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0043/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 44. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=1, vector=[6, 1, -15]. Give (scalar,vector,scaled vector).

View answer and explanation(1, [6, 1, -15], [6, 1, -15])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, [6, 1, -15], [6, 1, -15]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0044/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 45. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=2, vector=[7, 2, -16]. Give (scalar,vector,scaled vector).

View answer and explanation(2, [7, 2, -16], [14, 4, -32])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, [7, 2, -16], [14, 4, -32]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0045/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 46. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[8, -2, -17]. Give (scalar,vector,scaled vector).

View answer and explanation(3, [8, -2, -17], [24, -6, -51])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, [8, -2, -17], [24, -6, -51]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0046/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 47. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-3, vector=[9, -1, -18]. Give (scalar,vector,scaled vector).

View answer and explanation(-3, [9, -1, -18], [-27, 3, 54])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, [9, -1, -18], [-27, 3, 54]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0047/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 48. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-2, vector=[10, 0, -19]. Give (scalar,vector,scaled vector).

View answer and explanation(-2, [10, 0, -19], [-20, 0, 38])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, [10, 0, -19], [-20, 0, 38]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0048/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 49. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=-1, vector=[11, 1, -20]. Give (scalar,vector,scaled vector).

View answer and explanation(-1, [11, 1, -20], [-11, -1, 20])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, [11, 1, -20], [-11, -1, 20]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0049/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 50. A Declared Real-Scalar Instance on Integer Coordinates

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=0, vector=[12, 2, -21]. Give (scalar,vector,scaled vector).

View answer and explanation(0, [12, 2, -21], [0, 0, 0])

Multiply every coordinate by the same scalar; zero scalar gives the zero vector in the same dimension. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, [12, 2, -21], [0, 0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0050/)

Real vector space · qf-str22-vectorspace-scalar-multiplication · candidate





## 51. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-12, -2, 3]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-12, -2, 3], [12, 2, -3], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, -2, 3], [12, 2, -3], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0051/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 52. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-11, -1, 2]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-11, -1, 2], [11, 1, -2], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, -1, 2], [11, 1, -2], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0052/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 53. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-10, 0, 1]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-10, 0, 1], [10, 0, -1], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, 0, 1], [10, 0, -1], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0053/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 54. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-9, 1, 0]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-9, 1, 0], [9, -1, 0], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 1, 0], [9, -1, 0], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0054/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 55. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-8, 2, -1]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-8, 2, -1], [8, -2, 1], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 2, -1], [8, -2, 1], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0055/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 56. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-7, -2, -2]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-7, -2, -2], [7, 2, 2], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, -2, -2], [7, 2, 2], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0056/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 57. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-6, -1, -3]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-6, -1, -3], [6, 1, 3], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, -1, -3], [6, 1, 3], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0057/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 58. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-5, 0, -4]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-5, 0, -4], [5, 0, 4], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, 0, -4], [5, 0, 4], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0058/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 59. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-4, 1, -5]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-4, 1, -5], [4, -1, 5], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, 1, -5], [4, -1, 5], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0059/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 60. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-3, 2, -6]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-3, 2, -6], [3, -2, 6], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, 2, -6], [3, -2, 6], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0060/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 61. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-2, -2, -7]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-2, -2, -7], [2, 2, 7], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, -2, -7], [2, 2, 7], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0061/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 62. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-1, -1, -8]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([-1, -1, -8], [1, 1, 8], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, -1, -8], [1, 1, 8], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0062/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 63. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[0, 0, -9]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([0, 0, -9], [0, 0, 9], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 0, -9], [0, 0, 9], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0063/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 64. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[1, 1, -10]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([1, 1, -10], [-1, -1, 10], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 1, -10], [-1, -1, 10], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0064/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 65. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[2, 2, -11]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([2, 2, -11], [-2, -2, 11], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 2, -11], [-2, -2, 11], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0065/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 66. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[3, -2, -12]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([3, -2, -12], [-3, 2, 12], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, -2, -12], [-3, 2, 12], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0066/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 67. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[4, -1, -13]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([4, -1, -13], [-4, 1, 13], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, -1, -13], [-4, 1, 13], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0067/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 68. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[5, 0, -14]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([5, 0, -14], [-5, 0, 14], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0, -14], [-5, 0, 14], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0068/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 69. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[6, 1, -15]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([6, 1, -15], [-6, -1, 15], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 1, -15], [-6, -1, 15], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0069/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 70. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[7, 2, -16]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([7, 2, -16], [-7, -2, 16], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 2, -16], [-7, -2, 16], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0070/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 71. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[8, -2, -17]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([8, -2, -17], [-8, 2, 17], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, -2, -17], [-8, 2, 17], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0071/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 72. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[9, -1, -18]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([9, -1, -18], [-9, 1, 18], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, -1, -18], [-9, 1, 18], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0072/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 73. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[10, 0, -19]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([10, 0, -19], [-10, 0, 19], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 0, -19], [-10, 0, 19], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0073/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 74. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[11, 1, -20]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([11, 1, -20], [-11, -1, 20], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 1, -20], [-11, -1, 20], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0074/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 75. The Coordinate Additive Inverse and Its Zero Sum

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[12, 2, -21]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

View answer and explanation([12, 2, -21], [-12, -2, 21], [0, 0, 0], Yes)

Negate each coordinate and add in the same dimension. This verifies the supplied instance, not arbitrary abstract structures. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 2, -21], [-12, -2, 21], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0075/)

Real vector space · qf-str22-vectorspace-additive-inverse · candidate





## 76. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=-2, vector=[-12, -1, 2]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-3, -2, [-12, -1, 2], [60, 5, -10], [60, 5, -10], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, -2, [-12, -1, 2], [60, 5, -10], [60, 5, -10], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0076/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 77. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=-1, vector=[-11, 0, 1]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-2, -1, [-11, 0, 1], [33, 0, -3], [33, 0, -3], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, -1, [-11, 0, 1], [33, 0, -3], [33, 0, -3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0077/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 78. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=0, vector=[-10, 1, 0]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-1, 0, [-10, 1, 0], [10, -1, 0], [10, -1, 0], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, 0, [-10, 1, 0], [10, -1, 0], [10, -1, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0078/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 79. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=1, vector=[-9, 2, -1]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(0, 1, [-9, 2, -1], [-9, 2, -1], [-9, 2, -1], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, 1, [-9, 2, -1], [-9, 2, -1], [-9, 2, -1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0079/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 80. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=1, b=2, vector=[-8, -1, -2]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(1, 2, [-8, -1, -2], [-24, -3, -6], [-24, -3, -6], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, 2, [-8, -1, -2], [-24, -3, -6], [-24, -3, -6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0080/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 81. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=2, b=-2, vector=[-7, 0, -3]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(2, -2, [-7, 0, -3], [0, 0, 0], [0, 0, 0], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, -2, [-7, 0, -3], [0, 0, 0], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0081/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 82. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=3, b=-1, vector=[-6, 1, -4]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(3, -1, [-6, 1, -4], [-12, 2, -8], [-12, 2, -8], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, -1, [-6, 1, -4], [-12, 2, -8], [-12, 2, -8], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0082/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 83. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=0, vector=[-5, 2, -5]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-3, 0, [-5, 2, -5], [15, -6, 15], [15, -6, 15], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, 0, [-5, 2, -5], [15, -6, 15], [15, -6, 15], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0083/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 84. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=1, vector=[-4, -1, -6]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-2, 1, [-4, -1, -6], [4, 1, 6], [4, 1, 6], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, 1, [-4, -1, -6], [4, 1, 6], [4, 1, 6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0084/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 85. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=2, vector=[-3, 0, -7]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-1, 2, [-3, 0, -7], [-3, 0, -7], [-3, 0, -7], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, 2, [-3, 0, -7], [-3, 0, -7], [-3, 0, -7], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0085/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 86. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=-2, vector=[-2, 1, -8]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(0, -2, [-2, 1, -8], [4, -2, 16], [4, -2, 16], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, -2, [-2, 1, -8], [4, -2, 16], [4, -2, 16], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0086/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 87. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=1, b=-1, vector=[-1, 2, -9]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(1, -1, [-1, 2, -9], [0, 0, 0], [0, 0, 0], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, -1, [-1, 2, -9], [0, 0, 0], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0087/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 88. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=2, b=0, vector=[0, -1, -10]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(2, 0, [0, -1, -10], [0, -2, -20], [0, -2, -20], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, 0, [0, -1, -10], [0, -2, -20], [0, -2, -20], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0088/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 89. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=3, b=1, vector=[1, 0, -11]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(3, 1, [1, 0, -11], [4, 0, -44], [4, 0, -44], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, 1, [1, 0, -11], [4, 0, -44], [4, 0, -44], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0089/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 90. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=2, vector=[2, 1, -12]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-3, 2, [2, 1, -12], [-2, -1, 12], [-2, -1, 12], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, 2, [2, 1, -12], [-2, -1, 12], [-2, -1, 12], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0090/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 91. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=-2, vector=[3, 2, -13]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-2, -2, [3, 2, -13], [-12, -8, 52], [-12, -8, 52], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, -2, [3, 2, -13], [-12, -8, 52], [-12, -8, 52], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0091/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 92. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=-1, vector=[4, -1, -14]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-1, -1, [4, -1, -14], [-8, 2, 28], [-8, 2, 28], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, -1, [4, -1, -14], [-8, 2, 28], [-8, 2, 28], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0092/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 93. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=0, vector=[5, 0, -15]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(0, 0, [5, 0, -15], [0, 0, 0], [0, 0, 0], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, 0, [5, 0, -15], [0, 0, 0], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0093/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 94. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=1, b=1, vector=[6, 1, -16]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(1, 1, [6, 1, -16], [12, 2, -32], [12, 2, -32], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (1, 1, [6, 1, -16], [12, 2, -32], [12, 2, -32], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0094/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 95. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=2, b=2, vector=[7, 2, -17]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(2, 2, [7, 2, -17], [28, 8, -68], [28, 8, -68], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (2, 2, [7, 2, -17], [28, 8, -68], [28, 8, -68], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0095/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 96. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=3, b=-2, vector=[8, -1, -18]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(3, -2, [8, -1, -18], [8, -1, -18], [8, -1, -18], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (3, -2, [8, -1, -18], [8, -1, -18], [8, -1, -18], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0096/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 97. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-3, b=-1, vector=[9, 0, -19]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-3, -1, [9, 0, -19], [-36, 0, 76], [-36, 0, 76], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-3, -1, [9, 0, -19], [-36, 0, 76], [-36, 0, 76], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0097/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 98. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-2, b=0, vector=[10, 1, -20]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-2, 0, [10, 1, -20], [-20, -2, 40], [-20, -2, 40], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-2, 0, [10, 1, -20], [-20, -2, 40], [-20, -2, 40], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0098/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 99. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=-1, b=1, vector=[11, 2, -21]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(-1, 1, [11, 2, -21], [0, 0, 0], [0, 0, 0], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (-1, 1, [11, 2, -21], [0, 0, 0], [0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0099/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate





## 100. One Exact Instance of Distributivity over Scalar Addition

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=2, vector=[12, -1, -22]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?).

View answer and explanation(0, 2, [12, -1, -22], [24, -2, -44], [24, -2, -44], Yes)

Compute both coordinate lists independently. A general vector-space axiom would quantify over all relevant elements and scalars. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (0, 2, [12, -1, -22], [24, -2, -44], [24, -2, -44], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r04-real-vector-space-0100/)

Real vector space · qf-str22-vectorspace-scalar-distributivity · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261022-r04-real-vector-space-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261022-r04-real-vector-space-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261022-r04-real-vector-space-foundations/validation-report.json)