3.9 Matrix transformations questions for IB Maths AI

Find $w$ when $z=2+4\mathrm{i}$.

Describe fully the two transformations that map $z$ to $w$, stating the order in which they are applied.

Find the point $z$ that is mapped to $w=5-\mathrm{i}$.

The sprite is mapped by $z\mapsto \mathrm{i}z$. Find the images of $A$ and $C$.

Describe fully the single transformation $z\mapsto \mathrm{i}z$.

By writing $z=x+y\mathrm{i}$, show that the transformation $z\mapsto (a+b\mathrm{i})z$, where $a,b\in \mathbb{R}$, can be represented by the matrix

The sprite is now mapped by $z\mapsto (1+\mathrm{i})z$. Find the image of $C$.

Find the area of triangle $ABC$ and hence find the area of its image under $z\mapsto (1+\mathrm{i})z$.

Mei wants a single transformation $z\mapsto wz$ that maps $A$ onto $B$. Find $w$ in the form $a+b\mathrm{i}$.

(i) Find the image of $C$ under the transformation in part (f).

(ii) Hence state whether the image of the sprite is congruent to triangle $ABC$. Give a reason for your answer.

The matrix $\mathit{A}$ can be written as $k\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$, where $k>0$ and $0\mathrm{°}<\theta <90\mathrm{°}$.

(i) Find the value of $k$ and the value of $\theta$.

(ii) Hence describe fully the transformation represented by $\mathit{A}$.

Find the coordinates of the images under $T$ of the points $(8,0)$ and $(8,8)$.

Find $\det \mathit{A}$, and hence find the area of $S_1$.

Amara imagines continuing the pattern indefinitely. Show that the total area of all the squares $S_0,S_1,S_2,\dots$ is finite, and find this total area.

The point $C$ is the invariant point of $T$, so that $T$ maps $C$ to itself.

Find the coordinates of $C$.

Let $\mathit{c}$ be the position vector of $C$. Show that, for every position vector $\mathit{x}$, $T(\mathit{x})-\mathit{c}=\mathit{A}(\mathit{x}-\mathit{c})$.

Hence show that the distance $C{P}_n={0.75}^n\times C{P}_0$.

Amara draws the squares $S_0,S_1,S_2,\dots$ only while the vertex $P_n$ is at least 0.2 cm from $C$.

Find the number of squares in her logo.

Amara joins $P_0$ to $P_1$, $P_1$ to $P_2$, and so on, with straight line segments.

(i) Show that $P_{n+1}-P_n$, as a vector, is ${\mathit{A}}^n\left(\begin{array}{c}8\\ 0\end{array}\right)$.

(ii) Hence find the total length of this path if it were continued indefinitely.

Amara's colleague, Femi, suggests replacing $\mathit{A}$ with $\mathit{B}=\left(\begin{array}{cc}0.45& 0.6\\ 0.6& -0.45\end{array}\right)$.

The matrix $\mathit{B}$ represents a reflection in the line $y=(\tan \alpha )x$ combined with an enlargement, centred at the origin, with scale factor 0.75.

Find the equation of the line of reflection.

Find ${\mathit{B}}^2$. Hence explain why Femi's design would not produce a spiral of squares.

Describe fully the geometric effect of multiplying by $w$.

Find the exact value of $w$ in the form $a+b\mathrm{i}$.

The point $P$ represents $3-\mathrm{i}$. Find the exact image of $P$, giving your answer in the form $a+b\mathrm{i}$.

Write down the $2\times 2$ matrix that represents the same transformation.

Find the length of the segment $OE$ and the length of the segment $EW$.

Find the position vector of $G$.

A component sits at the point $K(50,20)$. Find the distance from $G$ to $K$.

The forearm, made up of segments $EW$ and $WG$, now rotates rigidly through $40\mathrm{°}$ anticlockwise about $E$. The shoulder segment $OE$ does not move.

Write down the matrix that represents an anticlockwise rotation of $40\mathrm{°}$ about the origin.

Find the new position vector of $G$.

Find the distance from the new position of $G$ to $K$.

The gripper then moves in a straight line directly towards $K$ at a constant speed of $1.5{\text{ cm s}}^{-1}$. Find its velocity vector.

Write down

(i) the matrix $\mathit{R}$ representing the rotation;

(ii) the matrix $\mathit{E}$ representing the enlargement.

Find the single matrix that represents the rotation followed by the enlargement.

The character starts at the point $(2,-5)$. Find its position after both transformations.