SOLUTION: Which matrix represents the rotation of the vector (1,4) by 2pi/3 radians? a. [-3.81, 1.57] b. [-3.96, -1.13] c. [-4, -1] d. [-3.74, -1.73] e. [-1, -4]

Instead of doing your problem for you, I'll do one exactly like it with
different numbers.

We draw the given vector:



We calculate the length (magnitude) of the original vector by the
Pythagorean theorem:



That'll also be the magnitude of the vector after we've rotate it. 

The angle that the original vector makes with the x-axis has tangent 5/1 or 5

Make sure your calculator is in radian mode, find

tan-1(5) = 1.373400767 radians

next we add  to that and get 3.258356359, which
is the direction angle of the rotated vector.

We sketch in the rotated vector:



We find the horizontal component of the rotated vector by multiplying
its magnitude by the cosine of its angle.



We find the vertical component of the rotated vector by multiplying
its magnitude by the sine of its angle.



Answer, rounded off:

[-5.06,-0.59]

Now do yours exactly the same way.

Edwin