Point A is at #(5 ,7 )# and point B is at #(-2 ,6 )#. 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 · 2017-05-16T18:16:45

The new coordinates are #=(7,-5)# and the distance has changed by #=7.14#

The matrix of a rotation clockwise by #1/2pi# about the origin is

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

Therefore, the trasformation of point #A# is

#A'=((0,1),(-1,0))((5),(7))=((7),(-5))#

Distance #AB# is

#=sqrt((-2-5)^2+(6-7)^2)#

#=sqrt(49+1)#

#=sqrt50#

Distance #A'B# is

#=sqrt((-2-7)^2+(6+5)^2)#

#=sqrt(81+121)#

#=sqrt202#

The distance has changed by

#=sqrt202-sqrt50#

#=7.14#