Point A is at #(-5 ,1 )# and point B is at #(2 ,-3 )#. 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 · 2017-06-02T11:40:11

The distance has changed by #=4.46#

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# into #A'# is

#A'=((0,-1),(1,0))((-5),(1))=((-1),(-5))#

Distance #AB# is

#=sqrt((2-(-5))^2+(-3-(-1))^2)#

#=sqrt(49+16)#

#=sqrt65#

Distance #A'B# is

#=sqrt((2-(-1))^2+(-3-(-5))^2)#

#=sqrt(9+4)#

#=sqrt13#

The distance has changed by

#=sqrt65-sqrt13#

#=4.46#