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?

1 Answer
Jan 26, 2016

Answer:

(-1, 2) ==> (1, -2)
(-4, 1) ==> (4, -1)
Something to note- Rotation by #pi# simply flips signs![enter image source here]

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)