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#