Which vector is not collinear with vector a=[6,-4]?

A: [3, -2]
B: [-6, -4]
C: [-6, 4]
D: [-9, 6]

To determine which vector is not collinear with vector a=[6,-4], we need to check if the vectors have the same or opposite directions. If they have the same or opposite directions, they are collinear.

Let's analyze each option:

A: [3, -2]

To check if it is collinear with vector a, we need to calculate the direction.

Direction of vector a: a = [6, -4]
Direction of vector A: A = [3, -2]

To calculate the direction, we divide each component of the vector by its magnitude.

Direction of vector a: [6/√(6^2+(-4)^2), -4/√(6^2+(-4)^2)] = [6/√(36+16), -4/√(36+16)] = [6/√(52), -4/√(52)] = [6/2√13, -4/2√13] = [3/√13, -2/√13]

Direction of vector A: [3/√(3^2+(-2)^2), -2/√(3^2+(-2)^2)] = [3/√(9+4), -2/√(9+4)] = [3/√(13), -2/√(13)]

The direction of vector a is [3/√13, -2/√13], and the direction of vector A is [3/√(13), -2/√(13)].

Since the directions are the same, vector A is collinear with vector a.

B: [-6, -4]

To check if it is collinear with vector a, we need to calculate the direction.

Direction of vector B: B = [-6/√(6^2+(-4)^2), -4/√(6^2+(-4)^2)] = [-6/√(36+16), -4/√(36+16)] = [-6/√(52), -4/√(52)] = [-6/2√13, -4/2√13] = [-3/√13, -2/√13]

The direction of vector B is [-3/√13, -2/√13].

Since the directions are the same, vector B is collinear with vector a.

C: [-6, 4]

To check if it is collinear with vector a, we need to calculate the direction.

Direction of vector C: C = [-6/√(6^2+4^2), 4/√(6^2+4^2)] = [-6/√(36+16), 4/√(36+16)] = [-6/√(52), 4/√(52)] = [-6/2√13, 4/2√13] = [-3/√13, 2/√13]

The direction of vector C is [-3/√13, 2/√13].

Since the directions are different, vector C is not collinear with vector a.

D: [-9, 6]

To check if it is collinear with vector a, we need to calculate the direction.

Direction of vector D: D = [-9/√(9^2+6^2), 6/√(9^2+6^2)] = [-9/√(81+36), 6/√(81+36)] = [-9/√(117), 6/√(117)] = [-9/√(9⋅13), 6/√(9⋅13)] = [-9/(3√13), 6/(3√13)] = [-3/√13, 2/√13]

The direction of vector D is [-3/√13, 2/√13].

Since the directions are different, vector D is not collinear with vector a.

Therefore, the vector that is not collinear with vector a=[6,-4] is option C: [-6, 4].

To determine which vector is not collinear with vector a=[6,-4], we need to check if any of the given vectors are scalar multiples of vector a. If a vector is a scalar multiple of vector a, then the two vectors are collinear.

To do this, we can calculate the scalar factor (r) between each of the given vectors and vector a. If the resulting scalar factor is the same for two vectors, then they are scalar multiples of each other and thus collinear with vector a.

Let's calculate the scalar factor (r) for each vector:

A: [3, -2]
To find r, we can divide the x-coordinate of vector A (-3) by the x-coordinate of vector a (6):
r = -3/6 = -1/2

B: [-6, -4]
To find r, we can divide the x-coordinate of vector B (-6) by the x-coordinate of vector a (6):
r = -6/6 = -1

C: [-6, 4]
To find r, we can divide the x-coordinate of vector C (-6) by the x-coordinate of vector a (6):
r = -6/6 = -1

D: [-9, 6]
To find r, we can divide the x-coordinate of vector D (-9) by the x-coordinate of vector a (6):
r = -9/6 = -3/2

Now, let's compare the scalar factors (r) we calculated:
-1/2, -1, -1, -3/2

None of the scalar factors are equal, which means none of the given vectors are collinear with vector a=[6,-4]. Therefore, there is no vector that is not collinear with vector a=[6,-4].

C and D are parallel to a [6,-4], but in the opposite direction. A is parallel. B is not parallel