Point A is at $(-3 ,4 )$ and point B is at $(-8 ,1 )$ . 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 · 2017-11-24T16:46:29

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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 A:

$A=((-3),(color(white)(8)4))$

Transformation matrix will be:

$((cos((3pi)/2),-sin((3pi)/2)),(sin((3pi)/2),color(white)(8)cos((3pi)/2)))$
$:.$

$A'=((cos((3pi)/2),-sin((3pi)/2)),(sin((3pi)/2),color(white)(8)cos((3pi)/2)))((-3),(color(white)(8)4))=((4),(3))$

Coordinates:

$( 4 , 3 )$

Distance between A and B:

$d=sqrt((-3-(-8))^2+(4-1)^2)=sqrt(34)$

Distance between $A' and B$

$d=sqrt((4-(-8))^2+(3-1)^2)=4$

The distance has been reduced by a factor of $(2sqrt(34))/17$

$:.$

$(2sqrt(34))/17*sqrt(34)=4$

Point A is at (-3 ,4 )(-3 ,4 ) and point B is at (-8 ,1 )(-8 ,1 ). 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?