Write down `2xx2` matrix A which corresponds to a counterclockwise rotation of `60^(@)` about the origin. In the diagram OB of `2sqrt2` units in length. The square is rotated counterclockwise about O through `60^(@)` find the coordinates of the vertices of the square after rotating.

Explanation

the matrix desribes a rotation through an angle `60^(@)` in counterclockwise direction is

`[(cos60^(@),-sin60^(@)),(sin60^(@),cos60^(@))]=[((1)/(2),(-sqrt3)/(2)),((-sqrt3)/(2),(1)/(2))]=(1)/(2)[(1,-sqrt3),(sqrt3,1)]`
Since, each side of the square be x,
then `x^(2)+x^(2)=(2sqrt2)^(2)`
`rArr" " 2x^(2)=8 rArr x^(2) =4`
`therefore " " x=2 units`
therefore the coordinates of the vertices O,A,B and C are `(0,0),(2,0),`
`(2,2) and (0,2),` respectively. Let after rotation A map into A,B map into B,C map into C but the O map into itself.
If coordinates of A'B' and C' are `(x',y),(x'',y'') and (x''',y'''),` respectively.
`therefore" " [(,x'),(,y')]=(1)/(2)[(1,-sqrt3),(sqrt3,1)][(,2),(,0)]=(1)/(2)[(,2),(,2sqrt3)]=[(,1),(,sqrt3)]`
`therefore" "x'=1,y'=sqrt3 rArr A(2,0) to A'(1,sqrt3)`
and `[(,x''),(,y'')]=(1)/(2)[(1,-sqrt3),(sqrt3,1)][(,2),(,2)]=(1)/(2)[(2-2sqrt3),(2sqrt3+2)]=[(1-sqrt3),(sqrt3+1)]`
`therefore " " x''=1-sqrt3,y''=sqrt3+1`
`rArr " " B(2,2) to B'(1-sqrt3,sqrt3+1)`
`[(,x''),(,y'')]=(1)/(2)[(1,-sqrt3),(sqrt3,1)][(,0),(,2)]=(1)/(2)[(-2sqrt3),(2)]=[(-sqrt3),(1)]`
`therefore " " x'''=-sqrt3,y'''=1`
`rArr" " C(0,2) to C'(-sqrt3,1)`