Question

A line segment with endpoints at `(-1 , 2 )` and `(-4, 1 )` is rotated clockwise by `pi`. What are the new endpoints of the line segment?

Answer 1

(-1, 2) ==> (1, -2)
(-4, 1) ==> (4, -1)
Something to note- Rotation by `pi` simply flips signs

Explanation:

use the rotation transformation matrix

`V_i = sumR_(ij) V'_j`
`(V_1, V_2, V_3) = (x, y, z)`
Where
`V_i " "`represented the vector, 1st vector below"
`R_(ij)" "` the Rotation Matrix"
`V_i " "`represented the vector, 2nd vector"

`|x'| = |costheta -sintheta " " 0| |x|`
`|y'| = |costheta -sintheta " " 0| |y|`
`|z'| = |0" " 0" " 1""| |z|`

Now (-1, 2, 0):
`|x'| = |-1" " 0" " 0|" "|-1|`
`|y'| = | 0" " -1" " 0|" "| 2|`
`|z'| = | 0" " 0" " 1|" "| 0|`
So (1, 2, 0)
and using the same (4, -1, 0)