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