Point A is at $(5 ,-2 )$ and point B is at $(-2 ,5 )$ . 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 · 2017-11-24T08:12:57

See below.

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It can be seen from the diagram, that a rotation about the origin through an angle $theta$ can be represented as:

$((1),(0))->((costheta),(sintheta))$ and $((0),(1))->((-sintheta),(color(white)(8)costheta))$

So the transformation matrix will be:

$((costheta,-sintheta),(sintheta,color(white)(8)costheta))$

Matrix for point A:

$A= ((5),(-2))$

Transformation:- Rotation through $pi/2$ clockwise. This is equivalent to $2pi-pi/2=(3pi)/2$ anticlocwise.

$((cos((3pi)/2),-sin((3pi)/2)),(sin((3pi)/2),color(white)(8)cos((3pi)/2))) ((color(white)(88)5),(-2))=((-2),(color(white)()-5))$

$A'=((-2),(color(white)()-5))$

Distance between A and B:

$d=sqrt((5-(-2))^2+(-2-(-5))^2)=7sqrt2$

Distance between A' and B:

$d=sqrt((-2-(-2))^2+(-5-5)^2)=10$

The distance between the points has increased by a factor of $(5sqrt(2))/7$

Point A is at (5 ,-2 )(5 ,-2 ) and point B is at (-2 ,5 )(-2 ,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?