# Plane Vectors in a Fixed Basis

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-20261003-advance-20261004-r05-vector-foundations · 2026-10-03 · 100 questions

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



## 1. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(0, 0) and v=(0, 0). Compute the exact components of u+v.

View answer and explanation(0, 0)

Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0001/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 2. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-11, -1) and v=(11, -2). Compute the exact components of u+v.

View answer and explanation(0, -3)

Apply addition componentwise to obtain (0, -3). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0002/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 3. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-10, 0) and v=(10, -1). Compute the exact components of u+v.

View answer and explanation(0, -1)

Apply addition componentwise to obtain (0, -1). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0003/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 4. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-9, 1) and v=(9, 0). Compute the exact components of u+v.

View answer and explanation(0, 1)

Apply addition componentwise to obtain (0, 1). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0004/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 5. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-8, 2) and v=(8, 1). Compute the exact components of u+v.

View answer and explanation(0, 3)

Apply addition componentwise to obtain (0, 3). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0005/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 6. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-7, -2) and v=(7, 2). Compute the exact components of u+v.

View answer and explanation(0, 0)

Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0006/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 7. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-6, -1) and v=(6, 3). Compute the exact components of u+v.

View answer and explanation(0, 2)

Apply addition componentwise to obtain (0, 2). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0007/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 8. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-5, 0) and v=(5, -3). Compute the exact components of u+v.

View answer and explanation(0, -3)

Apply addition componentwise to obtain (0, -3). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0008/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 9. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-4, 1) and v=(4, -2). Compute the exact components of u+v.

View answer and explanation(0, -1)

Apply addition componentwise to obtain (0, -1). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0009/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 10. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-3, 2) and v=(3, -1). Compute the exact components of u+v.

View answer and explanation(0, 1)

Apply addition componentwise to obtain (0, 1). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0010/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 11. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-2, -2) and v=(2, 0). Compute the exact components of u+v.

View answer and explanation(0, -2)

Apply addition componentwise to obtain (0, -2). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0011/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 12. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(-1, -1) and v=(1, 1). Compute the exact components of u+v.

View answer and explanation(0, 0)

Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0012/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 13. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(0, 0) and v=(0, 2). Compute the exact components of u+v.

View answer and explanation(0, 2)

Apply addition componentwise to obtain (0, 2). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0013/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 14. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(1, 1) and v=(-1, 3). Compute the exact components of u+v.

View answer and explanation(0, 4)

Apply addition componentwise to obtain (0, 4). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0014/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 15. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(2, 2) and v=(-2, -3). Compute the exact components of u+v.

View answer and explanation(0, -1)

Apply addition componentwise to obtain (0, -1). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0015/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 16. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(3, -2) and v=(-3, -2). Compute the exact components of u+v.

View answer and explanation(0, -4)

Apply addition componentwise to obtain (0, -4). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0016/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 17. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(4, -1) and v=(-4, -1). Compute the exact components of u+v.

View answer and explanation(0, -2)

Apply addition componentwise to obtain (0, -2). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0017/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 18. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(5, 0) and v=(-5, 0). Compute the exact components of u+v.

View answer and explanation(0, 0)

Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0018/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 19. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(6, 1) and v=(-6, 1). Compute the exact components of u+v.

View answer and explanation(0, 2)

Apply addition componentwise to obtain (0, 2). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0019/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 20. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(7, 2) and v=(-7, 2). Compute the exact components of u+v.

View answer and explanation(0, 4)

Apply addition componentwise to obtain (0, 4). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0020/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 21. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(8, -2) and v=(-8, 3). Compute the exact components of u+v.

View answer and explanation(0, 1)

Apply addition componentwise to obtain (0, 1). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0021/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 22. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(9, -1) and v=(-9, -3). Compute the exact components of u+v.

View answer and explanation(0, -4)

Apply addition componentwise to obtain (0, -4). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0022/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 23. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(10, 0) and v=(-10, -2). Compute the exact components of u+v.

View answer and explanation(0, -2)

Apply addition componentwise to obtain (0, -2). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0023/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 24. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(11, 1) and v=(-11, -1). Compute the exact components of u+v.

View answer and explanation(0, 0)

Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0024/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 25. Componentwise Vector Addition in a Fixed Basis

In the fixed basis of the real plane R², u=(12, 2) and v=(-12, 0). Compute the exact components of u+v.

View answer and explanation(0, 2)

Apply addition componentwise to obtain (0, 2). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0025/)

Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate





## 26. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(0, 0) and scalar k=0. Compute the exact components of ku.

View answer and explanation(0, 0)

Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0026/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 27. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-11, -1) and scalar k=-2. Compute the exact components of ku.

View answer and explanation(22, 2)

Multiply both components by k to obtain ku=(22, 2). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0027/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 28. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-10, 0) and scalar k=-1. Compute the exact components of ku.

View answer and explanation(10, 0)

Multiply both components by k to obtain ku=(10, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0028/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 29. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-9, 1) and scalar k=0. Compute the exact components of ku.

View answer and explanation(0, 0)

Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0029/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 30. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-8, 2) and scalar k=1. Compute the exact components of ku.

View answer and explanation(-8, 2)

Multiply both components by k to obtain ku=(-8, 2). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0030/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 31. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-7, -2) and scalar k=2. Compute the exact components of ku.

View answer and explanation(-14, -4)

Multiply both components by k to obtain ku=(-14, -4). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0031/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 32. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-6, -1) and scalar k=3. Compute the exact components of ku.

View answer and explanation(-18, -3)

Multiply both components by k to obtain ku=(-18, -3). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0032/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 33. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-5, 0) and scalar k=-3. Compute the exact components of ku.

View answer and explanation(15, 0)

Multiply both components by k to obtain ku=(15, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0033/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 34. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-4, 1) and scalar k=-2. Compute the exact components of ku.

View answer and explanation(8, -2)

Multiply both components by k to obtain ku=(8, -2). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0034/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 35. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-3, 2) and scalar k=-1. Compute the exact components of ku.

View answer and explanation(3, -2)

Multiply both components by k to obtain ku=(3, -2). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0035/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 36. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-2, -2) and scalar k=0. Compute the exact components of ku.

View answer and explanation(0, 0)

Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0036/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 37. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(-1, -1) and scalar k=1. Compute the exact components of ku.

View answer and explanation(-1, -1)

Multiply both components by k to obtain ku=(-1, -1). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0037/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 38. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(0, 0) and scalar k=2. Compute the exact components of ku.

View answer and explanation(0, 0)

Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0038/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 39. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(1, 1) and scalar k=3. Compute the exact components of ku.

View answer and explanation(3, 3)

Multiply both components by k to obtain ku=(3, 3). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0039/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 40. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(2, 2) and scalar k=-3. Compute the exact components of ku.

View answer and explanation(-6, -6)

Multiply both components by k to obtain ku=(-6, -6). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0040/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 41. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(3, -2) and scalar k=-2. Compute the exact components of ku.

View answer and explanation(-6, 4)

Multiply both components by k to obtain ku=(-6, 4). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0041/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 42. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(4, -1) and scalar k=-1. Compute the exact components of ku.

View answer and explanation(-4, 1)

Multiply both components by k to obtain ku=(-4, 1). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0042/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 43. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(5, 0) and scalar k=0. Compute the exact components of ku.

View answer and explanation(0, 0)

Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0043/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 44. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(6, 1) and scalar k=1. Compute the exact components of ku.

View answer and explanation(6, 1)

Multiply both components by k to obtain ku=(6, 1). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0044/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 45. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(7, 2) and scalar k=2. Compute the exact components of ku.

View answer and explanation(14, 4)

Multiply both components by k to obtain ku=(14, 4). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0045/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 46. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(8, -2) and scalar k=3. Compute the exact components of ku.

View answer and explanation(24, -6)

Multiply both components by k to obtain ku=(24, -6). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0046/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 47. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(9, -1) and scalar k=-3. Compute the exact components of ku.

View answer and explanation(-27, 3)

Multiply both components by k to obtain ku=(-27, 3). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0047/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 48. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(10, 0) and scalar k=-2. Compute the exact components of ku.

View answer and explanation(-20, 0)

Multiply both components by k to obtain ku=(-20, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0048/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 49. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(11, 1) and scalar k=-1. Compute the exact components of ku.

View answer and explanation(-11, -1)

Multiply both components by k to obtain ku=(-11, -1). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0049/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 50. Scalar Multiplication in a Fixed Basis

In the fixed basis of the real plane R², u=(12, 2) and scalar k=0. Compute the exact components of ku.

View answer and explanation(0, 0)

Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0050/)

Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate





## 51. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-12, -2) and v=(-3, 12). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-9, -14)

Apply ordered subtraction componentwise to obtain (-9, -14). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0051/)

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





## 52. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-11, -1) and v=(-2, 11). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-9, -12)

Apply ordered subtraction componentwise to obtain (-9, -12). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0052/)

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





## 53. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-10, 0) and v=(-1, 10). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-9, -10)

Apply ordered subtraction componentwise to obtain (-9, -10). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0053/)

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





## 54. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-9, 1) and v=(0, 9). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-9, -8)

Apply ordered subtraction componentwise to obtain (-9, -8). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0054/)

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





## 55. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-8, 2) and v=(1, 8). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-9, -6)

Apply ordered subtraction componentwise to obtain (-9, -6). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0055/)

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





## 56. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-7, -2) and v=(2, 7). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-9, -9)

Apply ordered subtraction componentwise to obtain (-9, -9). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0056/)

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





## 57. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-6, -1) and v=(3, 6). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-9, -7)

Apply ordered subtraction componentwise to obtain (-9, -7). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0057/)

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





## 58. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-5, 0) and v=(-3, 5). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-2, -5)

Apply ordered subtraction componentwise to obtain (-2, -5). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0058/)

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





## 59. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-4, 1) and v=(-2, 4). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-2, -3)

Apply ordered subtraction componentwise to obtain (-2, -3). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0059/)

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





## 60. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-3, 2) and v=(-1, 3). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-2, -1)

Apply ordered subtraction componentwise to obtain (-2, -1). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0060/)

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





## 61. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-2, -2) and v=(0, 2). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-2, -4)

Apply ordered subtraction componentwise to obtain (-2, -4). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0061/)

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





## 62. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(-1, -1) and v=(1, 1). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-2, -2)

Apply ordered subtraction componentwise to obtain (-2, -2). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0062/)

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





## 63. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(0, 0) and v=(2, 0). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-2, 0)

Apply ordered subtraction componentwise to obtain (-2, 0). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0063/)

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





## 64. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(1, 1) and v=(3, -1). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(-2, 2)

Apply ordered subtraction componentwise to obtain (-2, 2). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0064/)

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





## 65. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(2, 2) and v=(-3, -2). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(5, 4)

Apply ordered subtraction componentwise to obtain (5, 4). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0065/)

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





## 66. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(3, -2) and v=(-2, -3). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(5, 1)

Apply ordered subtraction componentwise to obtain (5, 1). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0066/)

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





## 67. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(4, -1) and v=(-1, -4). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(5, 3)

Apply ordered subtraction componentwise to obtain (5, 3). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0067/)

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





## 68. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(5, 0) and v=(0, -5). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(5, 5)

Apply ordered subtraction componentwise to obtain (5, 5). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0068/)

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





## 69. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(6, 1) and v=(1, -6). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(5, 7)

Apply ordered subtraction componentwise to obtain (5, 7). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0069/)

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





## 70. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(7, 2) and v=(2, -7). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(5, 9)

Apply ordered subtraction componentwise to obtain (5, 9). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0070/)

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





## 71. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(8, -2) and v=(3, -8). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(5, 6)

Apply ordered subtraction componentwise to obtain (5, 6). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0071/)

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





## 72. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(9, -1) and v=(-3, -9). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(12, 8)

Apply ordered subtraction componentwise to obtain (12, 8). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0072/)

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





## 73. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(10, 0) and v=(-2, -10). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(12, 10)

Apply ordered subtraction componentwise to obtain (12, 10). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0073/)

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





## 74. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(11, 1) and v=(-1, -11). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(12, 12)

Apply ordered subtraction componentwise to obtain (12, 12). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0074/)

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





## 75. Ordered Vector Subtraction in a Fixed Basis

In the fixed basis of the real plane R², u=(12, 2) and v=(0, -12). Compute the exact components of u−v, preserving the subtraction order.

View answer and explanation(12, 14)

Apply ordered subtraction componentwise to obtain (12, 14). These are vector components, not a geometric point or a length.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0075/)

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





## 76. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(0, 0) and v=(0, 0). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0076/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 77. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-11, -1) and v=(-10, -2). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0077/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 78. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-10, 0) and v=(-10, 0). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0078/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 79. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-9, 1) and v=(1, -9). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0079/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 80. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-8, 2) and v=(-8, 2). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0080/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 81. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-7, -2) and v=(-6, -3). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0081/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 82. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-6, -1) and v=(-6, -1). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0082/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 83. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-5, 0) and v=(-4, -1). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0083/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 84. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-4, 1) and v=(-4, 1). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0084/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 85. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-3, 2) and v=(-2, 1). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0085/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 86. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-2, -2) and v=(-2, -2). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0086/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 87. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(-1, -1) and v=(0, -2). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0087/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 88. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(0, 4) and v=(0, 4). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0088/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 89. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(1, 1) and v=(2, 0). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0089/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 90. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(2, 2) and v=(2, 2). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0090/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 91. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(3, -2) and v=(4, -3). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0091/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 92. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(4, -1) and v=(4, -1). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0092/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 93. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(5, 0) and v=(6, -1). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0093/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 94. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(6, 1) and v=(6, 1). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0094/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 95. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(7, 2) and v=(8, 1). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0095/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 96. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(8, -2) and v=(8, -2). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0096/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 97. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(9, -1) and v=(10, -2). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0097/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 98. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(10, 0) and v=(10, 0). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0098/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 99. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(11, 1) and v=(12, 0). Are u and v equal?

View answer and explanationNo

Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0099/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate





## 100. Vector Equality in a Fixed Basis

In the fixed basis of the real plane R², u=(12, 2) and v=(12, 2). Are u and v equal?

View answer and explanationYes

Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r05-vector-0100/)

Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r05-vector-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261003-advance-20261004-r05-vector-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r05-vector-foundations/validation-report.json)