Point A is at #(1 ,-8 )# and point B is at #(-3 ,-2 )#. 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 · 2017-05-07T17:22:25

The new coordinates are #=(8,1)# and the change in the distance is #=4.2#

The matrix of a rotation clockwise by #3/2pi# about the origin is

#=((cos(-3/2pi),-sin(-3/2pi)),(sin(-3/2pi),cos(-3/2pi)))=((0,1),(-1,0))#

Therefore, the trasformation of point #A# is

#A'=((0,-1),(1,0))((1),(-8))=((8),(1))#

Distance #AB# is

#=sqrt((-3-1)^2+(-2+8)^2)#

#=sqrt(16+36)#

#=sqrt52#

Distance #A'B# is

#=sqrt((-3-8)^2+(-2-1)^2)#

#=sqrt(121+9)#

#=sqrt130#

The distance has changed by

#=sqrt130-sqrt52#

#=4.2#