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.
In the fixed basis of the real plane R², u=(0, 0) and v=(0, 0). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-11, -1) and v=(11, -2). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-10, 0) and v=(10, -1). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-9, 1) and v=(9, 0). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-8, 2) and v=(8, 1). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-7, -2) and v=(7, 2). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-6, -1) and v=(6, 3). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-5, 0) and v=(5, -3). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-4, 1) and v=(4, -2). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-3, 2) and v=(3, -1). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-2, -2) and v=(2, 0). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(-1, -1) and v=(1, 1). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(0, 0) and v=(0, 2). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(1, 1) and v=(-1, 3). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(2, 2) and v=(-2, -3). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(3, -2) and v=(-3, -2). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(4, -1) and v=(-4, -1). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(5, 0) and v=(-5, 0). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(6, 1) and v=(-6, 1). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(7, 2) and v=(-7, 2). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(8, -2) and v=(-8, 3). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(9, -1) and v=(-9, -3). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(10, 0) and v=(-10, -2). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(11, 1) and v=(-11, -1). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(12, 2) and v=(-12, 0). Compute the exact components of u+v.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-addition · candidate
In the fixed basis of the real plane R², u=(0, 0) and scalar k=0. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-11, -1) and scalar k=-2. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-10, 0) and scalar k=-1. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-9, 1) and scalar k=0. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-8, 2) and scalar k=1. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-7, -2) and scalar k=2. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-6, -1) and scalar k=3. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-5, 0) and scalar k=-3. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-4, 1) and scalar k=-2. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-3, 2) and scalar k=-1. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-2, -2) and scalar k=0. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(-1, -1) and scalar k=1. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(0, 0) and scalar k=2. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(1, 1) and scalar k=3. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(2, 2) and scalar k=-3. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(3, -2) and scalar k=-2. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(4, -1) and scalar k=-1. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(5, 0) and scalar k=0. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(6, 1) and scalar k=1. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(7, 2) and scalar k=2. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(8, -2) and scalar k=3. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(9, -1) and scalar k=-3. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(10, 0) and scalar k=-2. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(11, 1) and scalar k=-1. Compute the exact components of ku.
(-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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
In the fixed basis of the real plane R², u=(12, 2) and scalar k=0. Compute the exact components of ku.
(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
Plane vectors in a fixed basis · qf-vector-scalar-multiplication · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(-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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
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.
(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
Plane vectors in a fixed basis · qf-vector-coordinate-difference · candidate
In the fixed basis of the real plane R², u=(0, 0) and v=(0, 0). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-11, -1) and v=(-10, -2). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-10, 0) and v=(-10, 0). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-9, 1) and v=(1, -9). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-8, 2) and v=(-8, 2). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-7, -2) and v=(-6, -3). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-6, -1) and v=(-6, -1). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-5, 0) and v=(-4, -1). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-4, 1) and v=(-4, 1). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-3, 2) and v=(-2, 1). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-2, -2) and v=(-2, -2). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(-1, -1) and v=(0, -2). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(0, 4) and v=(0, 4). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(1, 1) and v=(2, 0). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(2, 2) and v=(2, 2). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(3, -2) and v=(4, -3). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(4, -1) and v=(4, -1). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(5, 0) and v=(6, -1). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(6, 1) and v=(6, 1). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(7, 2) and v=(8, 1). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(8, -2) and v=(8, -2). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(9, -1) and v=(10, -2). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(10, 0) and v=(10, 0). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(11, 1) and v=(12, 0). Are u and v equal?
No
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
In the fixed basis of the real plane R², u=(12, 2) and v=(12, 2). Are u and v equal?
Yes
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
Plane vectors in a fixed basis · qf-vector-coordinate-equality · candidate
Canonical question JSONL · English JSONL · Validation report