# Fixed-Basis Vector Combinations and Component Guards

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-20261012-advance-20261016-r01-vector-foundations · 2026-10-12 · 100 questions

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



## 1. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 0], v=[0, 0], a=0, b=0. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([0, 0], [0, 0], [0, 0])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0001/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 2. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[99, 1], v=[1, 2], a=0, b=1. Give (a*u, b*v, a*u+b*v) in that order.

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

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [1, 2], [1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0002/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 3. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 2], v=[1, 2], a=2, b=-2. Give (a*u, b*v, a*u+b*v) in that order.

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

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [-2, -4], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0003/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 4. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, -3], v=[1, 4], a=9007199254740991, b=-9007199254740991. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([18014398509481982, -27021597764222973], [-9007199254740991, -36028797018963964], [9007199254740991, -63050394783186937])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([18014398509481982, -27021597764222973], [-9007199254740991, -36028797018963964], [9007199254740991, -63050394783186937]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0004/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 5. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-8, 1], v=[1, 2], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([-16, 2], [-1, -2], [-17, 0])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-16, 2], [-1, -2], [-17, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0005/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 6. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 1], v=[1, 3], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([-14, 2], [-1, -3], [-15, -1])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-14, 2], [-1, -3], [-15, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0006/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 7. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-6, 1], v=[1, 4], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([-12, 2], [-1, -4], [-13, -2])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-12, 2], [-1, -4], [-13, -2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0007/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 8. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-5, 1], v=[1, 5], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

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

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10, 2], [-1, -5], [-11, -3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0008/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 9. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-4, 1], v=[1, 6], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([-8, 2], [-1, -6], [-9, -4])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 2], [-1, -6], [-9, -4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0009/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 10. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 1], v=[1, 7], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([-6, 2], [-1, -7], [-7, -5])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 2], [-1, -7], [-7, -5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0010/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 11. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-2, 1], v=[1, 8], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([-4, 2], [-1, -8], [-5, -6])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 2], [-1, -8], [-5, -6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0011/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 12. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-1, 1], v=[1, 9], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

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

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 2], [-1, -9], [-3, -7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0012/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 13. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 1], v=[1, 10], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([0, 2], [-1, -10], [-1, -8])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [-1, -10], [-1, -8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0013/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 14. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 1], v=[1, 11], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

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

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [-1, -11], [1, -9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0014/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 15. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, 1], v=[1, 12], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([4, 2], [-1, -12], [3, -10])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [-1, -12], [3, -10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0015/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 16. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[3, 1], v=[1, 13], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([6, 2], [-1, -13], [5, -11])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 2], [-1, -13], [5, -11]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0016/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 17. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[4, 1], v=[1, 14], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([8, 2], [-1, -14], [7, -12])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 2], [-1, -14], [7, -12]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0017/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 18. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 1], v=[1, 15], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([10, 2], [-1, -15], [9, -13])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 2], [-1, -15], [9, -13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0018/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 19. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[6, 1], v=[1, 16], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([12, 2], [-1, -16], [11, -14])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 2], [-1, -16], [11, -14]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0019/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 20. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 1], v=[1, 17], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([14, 2], [-1, -17], [13, -15])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([14, 2], [-1, -17], [13, -15]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0020/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 21. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[8, 1], v=[1, 18], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([16, 2], [-1, -18], [15, -16])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([16, 2], [-1, -18], [15, -16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0021/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 22. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[9, 1], v=[1, 19], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([18, 2], [-1, -19], [17, -17])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([18, 2], [-1, -19], [17, -17]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0022/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 23. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[10, 1], v=[1, 20], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([20, 2], [-1, -20], [19, -18])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([20, 2], [-1, -20], [19, -18]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0023/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 24. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[11, 1], v=[1, 21], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([22, 2], [-1, -21], [21, -19])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([22, 2], [-1, -21], [21, -19]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0024/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 25. Ordered Weighted Vector Combination

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[12, 1], v=[1, 22], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.

View answer and explanation([24, 2], [-1, -22], [23, -20])

Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([24, 2], [-1, -22], [23, -20]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0025/)

Plane vectors in a fixed basis · qf-density16-vector-weighted-combination · candidate





## 26. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 0], v=[0, 0]. Give (u, v, u−v, v−u, sum of the two differences).

View answer and explanation([0, 0], [0, 0], [0, 0], [0, 0], [0, 0])

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0], [0, 0], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0026/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 27. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, -1], v=[1, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, -1], [1, -1], [0, 0], [0, 0], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0027/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 28. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-10, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10, 1], [0, -1], [-10, 2], [10, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0028/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 29. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-9, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 1], [0, -1], [-9, 2], [9, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0029/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 30. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-8, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 1], [0, -1], [-8, 2], [8, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0030/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 31. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 1], [0, -1], [-7, 2], [7, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0031/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 32. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-6, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 1], [0, -1], [-6, 2], [6, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0032/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 33. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-5, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 1], [0, -1], [-5, 2], [5, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0033/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 34. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-4, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 1], [0, -1], [-4, 2], [4, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0034/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 35. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 1], [0, -1], [-3, 2], [3, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0035/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 36. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-2, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 1], [0, -1], [-2, 2], [2, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0036/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 37. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-1, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 1], [0, -1], [-1, 2], [1, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0037/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 38. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [0, -1], [0, 2], [0, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0038/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 39. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [0, -1], [1, 2], [-1, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0039/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 40. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1], [0, -1], [2, 2], [-2, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0040/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 41. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[3, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1], [0, -1], [3, 2], [-3, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0041/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 42. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[4, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 1], [0, -1], [4, 2], [-4, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0042/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 43. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 1], [0, -1], [5, 2], [-5, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0043/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 44. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[6, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 1], [0, -1], [6, 2], [-6, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0044/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 45. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 1], [0, -1], [7, 2], [-7, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0045/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 46. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[8, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 1], [0, -1], [8, 2], [-8, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0046/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 47. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[9, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 1], [0, -1], [9, 2], [-9, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0047/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 48. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[10, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 1], [0, -1], [10, 2], [-10, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0048/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 49. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[11, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 1], [0, -1], [11, 2], [-11, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0049/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 50. Both Directed Vector Differences and Their Sum

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[12, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).

View answer and explanation([12, 1], [0, -1], [12, 2], [-12, -2], [0, 0])

Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 1], [0, -1], [12, 2], [-12, -2], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0050/)

Plane vectors in a fixed basis · qf-density16-vector-difference-reversal · candidate





## 51. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 0], a=0, b=0. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([0, 0], [0, 0], [0, 0], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0051/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 52. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 2], a=2, b=-2. Give (u, (a+b)u, au+bu, difference of the last two routes).

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

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [0, 0], [0, 0], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0052/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 53. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, -1], a=-1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

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

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, -1], [1, -1], [1, -1], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0053/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 54. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-9, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([-9, 2], [-27, 6], [-27, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 2], [-27, 6], [-27, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0054/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 55. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-8, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([-8, 2], [-24, 6], [-24, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 2], [-24, 6], [-24, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0055/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 56. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([-7, 2], [-21, 6], [-21, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 2], [-21, 6], [-21, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0056/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 57. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-6, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([-6, 2], [-18, 6], [-18, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 2], [-18, 6], [-18, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0057/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 58. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-5, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

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

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 2], [-15, 6], [-15, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0058/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 59. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-4, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([-4, 2], [-12, 6], [-12, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 2], [-12, 6], [-12, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0059/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 60. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([-3, 2], [-9, 6], [-9, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 2], [-9, 6], [-9, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0060/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 61. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-2, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

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

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 2], [-6, 6], [-6, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0061/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 62. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-1, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

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

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 2], [-3, 6], [-3, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0062/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 63. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

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

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [0, 6], [0, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0063/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 64. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

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

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [3, 6], [3, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0064/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 65. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

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

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [6, 6], [6, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0065/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 66. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[3, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([3, 2], [9, 6], [9, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [9, 6], [9, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0066/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 67. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[4, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([4, 2], [12, 6], [12, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [12, 6], [12, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0067/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 68. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

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

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [15, 6], [15, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0068/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 69. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[6, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([6, 2], [18, 6], [18, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 2], [18, 6], [18, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0069/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 70. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([7, 2], [21, 6], [21, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 2], [21, 6], [21, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0070/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 71. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[8, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([8, 2], [24, 6], [24, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 2], [24, 6], [24, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0071/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 72. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[9, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([9, 2], [27, 6], [27, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 2], [27, 6], [27, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0072/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 73. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[10, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([10, 2], [30, 6], [30, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 2], [30, 6], [30, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0073/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 74. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[11, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([11, 2], [33, 6], [33, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 2], [33, 6], [33, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0074/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 75. Two Scalar-Addition Routes and Component Defect

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[12, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).

View answer and explanation([12, 2], [36, 6], [36, 6], [0, 0])

One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 2], [36, 6], [36, 6], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0075/)

Plane vectors in a fixed basis · qf-density16-vector-two-scalar-additivity · candidate





## 76. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 0], v=[0, 0], a=0, b=0, candidate=[0, 0]. Give (au+bv, candidate−actual value, is the candidate equal?).

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

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0076/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 77. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 0], v=[0, 1], a=1, b=1, candidate=[1, 0]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([1, 1], [0, -1], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [0, -1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0077/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 78. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 2], v=[1, 2], a=2, b=-1, candidate=[1, 2]. Give (au+bv, candidate−actual value, is the candidate equal?).

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

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0078/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 79. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-9, 1], v=[1, 2], a=2, b=1, candidate=[-17, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([-17, 4], [0, 0], Yes)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-17, 4], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0079/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 80. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-8, 1], v=[1, 2], a=2, b=1, candidate=[-14, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([-15, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-15, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0080/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 81. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 1], v=[1, 2], a=2, b=1, candidate=[-12, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([-13, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-13, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0081/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 82. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-6, 1], v=[1, 2], a=2, b=1, candidate=[-11, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

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

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-11, 4], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0082/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 83. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-5, 1], v=[1, 2], a=2, b=1, candidate=[-8, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([-9, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0083/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 84. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-4, 1], v=[1, 2], a=2, b=1, candidate=[-6, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([-7, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0084/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 85. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 1], v=[1, 2], a=2, b=1, candidate=[-5, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

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

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 4], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0085/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 86. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-2, 1], v=[1, 2], a=2, b=1, candidate=[-2, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([-3, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0086/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 87. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-1, 1], v=[1, 2], a=2, b=1, candidate=[0, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([-1, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0087/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 88. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 1], v=[1, 2], a=2, b=1, candidate=[1, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

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

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0088/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 89. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 1], v=[1, 2], a=2, b=1, candidate=[4, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([3, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0089/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 90. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, 1], v=[1, 2], a=2, b=1, candidate=[6, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([5, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0090/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 91. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[3, 1], v=[1, 2], a=2, b=1, candidate=[7, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([7, 4], [0, 0], Yes)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 4], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0091/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 92. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[4, 1], v=[1, 2], a=2, b=1, candidate=[10, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([9, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0092/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 93. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 1], v=[1, 2], a=2, b=1, candidate=[12, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([11, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0093/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 94. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[6, 1], v=[1, 2], a=2, b=1, candidate=[13, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

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

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([13, 4], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0094/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 95. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 1], v=[1, 2], a=2, b=1, candidate=[16, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([15, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([15, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0095/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 96. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[8, 1], v=[1, 2], a=2, b=1, candidate=[18, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([17, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([17, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0096/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 97. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[9, 1], v=[1, 2], a=2, b=1, candidate=[19, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([19, 4], [0, 0], Yes)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([19, 4], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0097/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 98. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[10, 1], v=[1, 2], a=2, b=1, candidate=[22, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([21, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([21, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0098/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 99. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[11, 1], v=[1, 2], a=2, b=1, candidate=[24, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([23, 4], [1, 0], No)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([23, 4], [1, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0099/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate





## 100. Weighted Vector Candidate and Ordered Residual

Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[12, 1], v=[1, 2], a=2, b=1, candidate=[25, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).

View answer and explanation([25, 4], [0, 0], Yes)

Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([25, 4], [0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r01-vector-0100/)

Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r01-vector-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261016-r01-vector-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r01-vector-foundations/validation-report.json)