Problems

01

Composing rotations

Write the matrix of rotation by $\theta$ and the matrix of rotation by $\phi$. Now apply the first matrix to $(x,y)$ and the second matrix to the output of $(x,y)$ under the action of the first matrix. Show that the resulting vector is the same as the vector given by applying a single rotation by the angle $\theta +\phi$. (You will need trig identities.)

02

Yaw and pitch do not commute

In this problem you will show that 3D rotations do not commute, meaning that the outcome of two rotations can be affected by the order in which the rotations are performed.

Write an arbitrary vector symbolically as $\left(\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right)$. Rotate this vector by a yaw of $+90^{\circ }$ using the yaw rotation matrix, and then rotate the output by a pitch of $+90^{\circ }$. Now do this in reverse: a pitch rotation of $+90^{\circ }$ followed by a yaw rotation of $+90^{\circ }$. Show that the results do not agree! Can you visualize this by rotating your hand?

03

Finding matrices

• (a) Find a $3\times 3$ matrix that acts as follows: $$\left(\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right)⟼\left(\begin{array}{c}3v_1-2v_3\\ 4v_1\\ v_1-v_2+v_3\end{array}\right).$$
• (b) Find a $3\times 2$ matrix that acts as follows: $$\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)⟼\left(\begin{array}{c}{v}_1-v_2\\ -2v_1+v_2\\ v_1\end{array}\right).$$
• (c) Find a $4\times 4$ matrix that acts as follows: $$\left(\begin{array}{c}{v}_1\\ v_2\\ v_3\\ v_4\end{array}\right)⟼\left(\begin{array}{c}{v}_1+v_2\\ v_2+v_3\\ v_3+v_4\\ 0\end{array}\right).$$
• (d) Find the matrix of a horizontal shear that maps ${\mathbf{e}}_2$ to ${\mathbf{e}}_2+3{\mathbf{e}}_1$ and does not change ${\mathbf{e}}_1$.

04

Rotations via reflections

Given an angle $\theta$ for a counterclockwise rotation of the $xy$-plane, find two lines such that reflecting in these two lines in sequence performs the rotation by $\theta$.

(This means that any rotation in the $xy$-plane can be accomplished by performing two reflections in sequence!)

Hint: try picking a very simple line as your first reflection line, e.g. an axis, and a more interesting line as your second reflection line.

You may find it convenient to consider that the formula

$$\left(\begin{array}{cc}1-2v_1{v}_1& -2v_1{v}_2\\ -2v_2{v}_1& 1-2v_2{v}_2\end{array}\right)$$

for a reflection matrix is satisfied by the matrix

$$\left(\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right).$$

(Expand the entries using double-angle formulas to see why!) This matrix gives the reflection through the line which meets the $x$-axis at the origin at an angle of $\theta$.