Point A is at #(3 ,9 )# 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 · 2017-05-29T10:30:18

The new coordinates of #A=(9,-3)# and the change in the distance is #=7.2#

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# into #A'# is

#A'=((0,1),(-1,0))((3),(9))=((9),(-3))#

Distance #AB# is

#=sqrt((-2-3)^2+(5-(9))^2)#

#=sqrt(25+16)#

#=sqrt41#

Distance #A'B# is

#=sqrt((-2-9)^2+(5-(-3))^2)#

#=sqrt(121+64)#

#=sqrt185#

The distance has changed by

#=sqrt185-sqrt41#

#=7.2#