Let be a point in the Cartesian plane

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Let

P(x, y)

be a point in the Cartesian plane. The point is rotated about the origin through an anticlockwise angle

\theta

to a new position

P'(x', y')

. (i) Derive expressions for

x'

and

y'

in terms of

x,\,y

and

\theta

. Your working should begin with the coordinates of

P

and the definition of rotation about the origin. (ii) Write the result of part (i) in the form of a

2\!\times\!2

matrix acting on the column vector

\begin{pmatrix}x\\y\end{pmatrix}

. State clearly the entries of the matrix.