Fixed-Basis Vector Combinations and Component Guards
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Plane vectors in a fixed basis · qf-density16-vector-candidate-residual · candidate
Canonical question JSONL · English JSONL · Validation report