Point A is at $(1 ,3 )$ and point B is at $(-7 ,-5 )$ . Point A is rotated $pi/2$ clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

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Answer 1 · 2018-05-16T17:12:48

$color(blue)((3,-1)$

$color(blue)(0.54337889 \ \ units)$

We can produce a rotation about the origin by using the transformation matrix:

$((cos(theta),-sin(theta)),(sin(theta),cos(theta)))$

This is for anticlockwise rotation, so for clockwise rotation we use the angle:

$2pi-pi/2=(3pi)/2$

So we have:

$((cos((3pi)/2),-sin((3pi)/2)),(sin((3pi)/2),cos((3pi)/2)))=((0,1),(-1,0))$

$A=(1,3)$

$A'=((0,1),(-1,0))((1),(3))=((3),(-1))$

Distance between A and B:

$d=sqrt((1-(-7))^2+(3-(-5))^2)=sqrt(128)=8sqrt(2)$

Distance between A' and B:

$d=sqrt((3-(-7))^2+(-1-(-5))^2)=sqrt(116)=2sqrt(29)$

Change in distance:

$8sqrt(2)-2sqrt(29)=0.54337889$ units

Point A is at (1 ,3 )(1 ,3 ) and point B is at (-7 ,-5 )(-7 ,-5 ). Point A is rotated pi/2 pi/2 clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?