## Name:

Due September 9

Use the questions below to identify any topics you should review. See the week 01 notes for background information on these topics.

## Trigonometry

Consider the point p and angle $\theta$ below, where p is a distance of 1 unit from the origin and $\theta$ is 45 degrees. What is the coordinate of p? Hint: what are the values of a and b in terms of $\theta$?

Consider the point $p_1$ and angle $\theta_1$ below. Suppose $p_1=(2,2,0)^T$. What is the value of $\theta_1$? Hint: Use tangent.

Consider the point $p_2$ and angle $\theta_2$ above. Suppose $p_2=(3,-2,0)^T$. What is the value of $\theta_2$? Hint: Use tangent.

## Vectors

A vector is an n-tuple of real numbers. In this class, we will work with 2D, 3D, and 4D vectors. Suppose we have a vector u=(-2, 3, 0)T and v=(-1, 4, 0)T.

• Draw the vectors u and v, with their tails anchored at the origin below.

• What is the length of u?

• What is the distance between u and v?

• Compute and draw u + v.

• Compute and draw u - v

• Compute the cross product $u \times v$.

• Normalize the vector u, e.g. compute $\frac{u}{\|u\|}$.

• Compute the dot product $u \cdot v$.

## Matrices

Consider the following matrices

$A=\begin{bmatrix} 1 & 3 \\ -0.5 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} -3 & 0 \\ 1 & 2 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 3 \\ -4 & 5 \\ 3 & -7 \end{bmatrix}$
• What are the dimensions of A, B, and C?

• What is the transpose of the matric C?

• Compute the products AB and BA.

• Is it possible to multiply C times itself? Why not? What about CCT?

• What is the product of $AA^{-1}$?

Consider the following matrix

$R = \begin{bmatrix} cos(30) & sin(30) & 0\\ -sin(30) & cos(30) & 0\\ 0 & 0 & 1 \end{bmatrix}$
• Suppose we have a vector u=(1,0,0)T. Draw u below. Then multiple u by R and draw Ru.

## Polynomials

Consider the polynomial $p(t) = 9t^3 + 6t^2$.

• What is the degree of $p(t)$?

• What is the derivative of $p(t)$?

• What is the value of $p(t)$ when t = -1?

Let $B_0(t) = (t - 1)^2$ and $B_1 = t - 2$.

• Compute an expression for $p(t) = B_0(t) + B_1(t)$ and re-arrange the terms into standard form

 Standard form has the following pattern: $a_nt^n + \ldots + a_2 t^2 + a t + a_0$.