Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.
Investigate the transfomations of the plane given by the 2 by 2
matrices with entries taking all combinations of values 0. -1 and
Explore the properties of matrix transformations with these 10 stimulating questions.
The point $P$ has coordinates $(r\cos \phi, r\sin \phi)$ where the distance of $P$ from the origin is $OP=r$ and the line $OP$ is at angle $\phi$ to the x axis. Find the image of this point under the transformation given by the 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 $y = x\tan\theta$. In the diagram below the lines $P'X'$ and $XA'$ are perpendicular to the line $OA'X'$. "
Prove that $OX = OX' = p$, $P'X' = PX = q$ and $OA = OA' = p\cos 2\theta.$ Find the lengths $BP',\ AX'$ and $BX'$ and hence prove that transformation given by the matrix
gives a reflection in the line $y=\tan\theta$.