Animation Math Review: Selfassessment
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 ntuple 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

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 CC^{T}?

What is the product of \(AA^{1}\)?
Consider the following matrix

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 rearrange the terms into standard form
Standard form has the following pattern: \(a_nt^n + \ldots + a_2 t^2 + a t + a_0\). 