Point A is at $(-5 ,-1 )$ and point B is at $(2 ,-3 )$ . Point A is rotated $(3pi)/2$ 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 · 2017-12-21T14:06:35

The new coordinates are $=(1,-5)$ and the distance has changed by $=5.04u$

The point $A=(-5,-1)$

The point $B=(2,-3)$

The distance

$AB=sqrt((2-(-5))^2+(-3-(-1))^2)=sqrt(49+4)=sqrt53$

The matrix for a rotation of angle $theta$ is

$r(theta)=((costheta,-sintheta),(sintheta,costheta))$

And when $theta=-3/2pi$

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

$=((0,-1),(1,0))$

Therefore,

The coordinates of the point $A'$ after the rotation of the point $A$ clockwise by $3/2pi$ is

$((x),(y))=((0,-1),(1,0))*((-5),(-1))=((1),(-5))$

The distance

$A'B=sqrt((2-1)^2+(-3-(-5))^2)=sqrt(1+4)=sqrt5$

The change in the distance is

$AB-A'B=sqrt53-sqrt5=5.04u$

Point A is at (-5 ,-1 )(-5 ,-1 ) and point B is at (2 ,-3 )(2 ,-3 ). Point A is rotated (3pi)/2 (3pi)/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?