# 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?

Jan 26, 2016

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

#### Explanation:

use the rotation transformation matrix

${V}_{i} = \sum {R}_{i j} V {'}_{j}$
$\left({V}_{1} , {V}_{2} , {V}_{3}\right) = \left(x , y , z\right)$
Where
${V}_{i} \text{ }$represented the vector, 1st vector below"
${R}_{i j} \text{ }$ the Rotation Matrix"
${V}_{i} \text{ }$represented the vector, 2nd vector"

$| x ' | = | \cos \theta - \sin \theta \text{ } 0 | | x |$
$| y ' | = | \cos \theta - \sin \theta \text{ } 0 | | y |$
$| z ' | = | 0 \text{ " 0" " 1} | | z |$

Now (-1, 2, 0):
$| x ' | = | - 1 \text{ " 0" " 0|" } | - 1 |$
$| y ' | = | 0 \text{ " -1" " 0|" } | 2 |$
$| z ' | = | 0 \text{ " 0" " 1|" } | 0 |$
So (1, 2, 0)
and using the same (4, -1, 0)