IB Maths AI HL 3.9 Matrix transformations and fractals

[Maximum number: 9]

François is a video game designer. He designs his games to take place in two dimensions, relative to an origin O . In one of his games, an object travels on a straight line $L_{1}$ with vector equation

r=\binom{-1}{1}+\lambda\binom{2}{-1}

(a)

Francois uses the matrix $\boldsymbol{T}=\begin{pmatrix}1 & 7 \\ 7 & -1\end{pmatrix}$ to transform $L_{1}$ into a new straight line $L_{2}$ . The object will then travel along $L_{2}$ .

Find the vector equation of $L_{2}$ .

[ 4 ]

\begin{pmatrix}1&7\\7&-1\end{pmatrix}\binom{-1+2\lambda}{1-\lambda}=\binom{6-5\lambda}{-8+15\lambda}

\mathbf{r}=\binom{6}{-8}+\lambda\binom{-5}{15}

(or equivalent)

Note: Award (M1) for the correct format of a vector equation of a line, A1 for the line being completely correct.

(b)

François knows that the transformation given by matrix $\boldsymbol{T}$ is made up of the following three separate transformations (in the order listed):
- A rotation of $\frac{\pi}{4}$ , anticlockwise (counter-clockwise) about the origin O
- An enlargement of scale factor $5\sqrt{2}$ , centred at O
- A reflection in the straight line y=mx, where $m=\tan \alpha, 0 \leq \alpha<\pi$

Write down the matrix that represents

[ 4 ]

(i)

the rotation.

[ 2 ]

\begin{pmatrix}\cos\left(\frac{\pi}{4}\right)&-\sin\left(\frac{\pi}{4}\right)\\\sin\left(\frac{\pi}{4}\right)&\cos\left(\frac{\pi}{4}\right)\end{pmatrix}

\begin{pmatrix}0.707&-0.707\\0.707&0.707\end{pmatrix}

\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix}

(ii)

the enlargement.

[ 2 ]

\begin{pmatrix}5\sqrt{2}&0\\0&5\sqrt{2}\end{pmatrix}

(c)

The matrix R represents the reflection. Write down R in terms of $\alpha$ .

[ 1 ]

\boldsymbol{R}=\begin{pmatrix}\cos 2\alpha&\sin 2\alpha\\\sin 2\alpha&-\cos 2\alpha\end{pmatrix}

IB Maths AI HL Ahl 3 9 Hl Matrix Transformations and Fractals Questions

Exam points

IB Maths AI HL Ahl 3 9 Hl Matrix Transformations and Fractals Questions question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)