Point A is at #(-3 ,4 )# and point B is at #(-8 ,1 )#. 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 · 2017-11-24T16:46:29

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It can be seen from the diagram, that a rotation about the origin through an angle #theta# can be represented as:

#((1),(0))->((costheta),(sintheta))# and #((0),(1))->((-sintheta),(color(white)(8)costheta))#

So the transformation matrix will be:

#((costheta,-sintheta),(sintheta,color(white)(8)costheta))#

Matrix A:

#A=((-3),(color(white)(8)4))#

Transformation matrix will be:

#((cos((3pi)/2),-sin((3pi)/2)),(sin((3pi)/2),color(white)(8)cos((3pi)/2)))#
#:.#

#A'=((cos((3pi)/2),-sin((3pi)/2)),(sin((3pi)/2),color(white)(8)cos((3pi)/2)))((-3),(color(white)(8)4))=((4),(3))#

Coordinates:

#( 4 , 3 )#

Distance between A and B:

#d=sqrt((-3-(-8))^2+(4-1)^2)=sqrt(34)#

Distance between #A' and B#

#d=sqrt((4-(-8))^2+(3-1)^2)=4#

The distance has been reduced by a factor of #(2sqrt(34))/17#

#:.#

#(2sqrt(34))/17*sqrt(34)=4#