The point

P

has coordinates

(r \cos ϕ, r \sin ϕ)

where the distance of

P

from the origin is

OP = r

and the line

OP

is at angle

ϕ

to the x axis. Find the image of this point under the transformation given by the matrix

T_{1} = (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix})

Draw a diagram and describe the effect of this transformation on the points of the plane.

The point $P$ has coordinates $(p, q)$ and the point $P'$ is the reflection of $P$ in the line $y = x \tan θ$ . In the diagram below the lines $P' X'$ and $XA'$ are perpendicular to the line $OA' X'$ . "

Reflection

Prove that

OX = OX' = p

P' X' = PX = q

and

OA = OA' = p \cos 2 θ .

Find the lengths

BP', AX'

and

BX'

and hence prove that transformation given by the matrix

T_{2} = (\begin{matrix} \cos 2 θ & \sin 2 θ \\ \sin 2 θ & - \cos 2 θ \end{matrix})

gives a reflection in the line

y = \tan θ