Show by using matrix method that a reflection about the line y=x followed by rotation about origin through 90° +ve is equivalent to reflection about y-axis.?

1 Answer
May 22, 2018

See below

Explanation:

Reflection about the line y = x

The effect of this reflection is to switch the x and y values of the reflected point. The matrix is:

  • A = ((0,1),(1,0))

CCW rotation of a point

For CCW rotations about origin by angle alpha:

  • R(alpha) = ((cos alpha, - sin alpha),(sin alpha , cos alpha))

If we combine these in the order suggested:

bb x' =A \ R(90^o) \ bb x

bb x' = ((0,1),(1,0)) ((0, - 1),(1 , 0)) bb x

= ((1,0),(0,-1)) bb x

implies ((x'),(y')) =((1,0),(0,-1)) ((x),(y)) = ((x),(-y))

That is equivalent to a reflection in x-axis .


Making it a CW rotation:

((x'),(y')) = ((0,1),(1,0)) ((0, 1),(-1 , 0))((x),(y))

= ((-1,0),(0,1)) ((x),(y)) = ((-x),(y))

That is a reflection in the y-axis.