Find the matrix that performs this translation.

Question

This problem will walk through the process of rotating by 90 degrees counterclockwise around the point (3, 5). Our strategy for rotation around the point (3, 5) will be to begin with a vector whose tail is at (3, 5), translate the vector so its tail is at the origin, perform the 90 degrees rotation, and then translate back to the point (3, 5). All matrices in this problem have the form:

a b c

d e f

0 0 1

(a) The first operation is a translation by (-3, -5), to move from (3, 5) back to the origin. Find the matrix (in the above form) that performs this translation.

(b) The second operation is a 90 degrees rotation counterclockwise around the origin. Find the matrix (in the above form) that performs this rotation.

(c) The third operation is a translation by (3, 5), to move back from the origin to the point (3, 5). Find the matrix (in the above form) that performs this translation.

(d) Use these matrices to find the matrix that performs a 90 degrees counterclockwise rotation around the point (3, 5).

(e) Use your matrix from (d) to determine where the point (-10, 5) ends up if it is rotated by 90 degrees around the point (3, 5).

Answer 1

a) + -3 -5

b) x 0 1

-1 0

c) + 3 5

d)combined result

x y + -3 -5 = x - 3 y - 5

x - 3 y - 5 * 0 1 = 5 - y x - 3

-1 0

5 - y + 3 = 8 - y

x - 3 5 2 + x

e) 8 - 5 = 3

2 + -10 -8

To see that this answer makes sense: (-10, 5) is 13 to the left of (3, 5), so if you rotate 90 degrees counterclockwise, you should end up 13 below (3, 5) (3, 5) + (0, -13) = (3, -8)

This is exactly where we end up.