1. Vectors and angles
Table of contents
Vectors and the dot product
Practice Questions 1.1-1.3
Question 1.1
Consider the vectors \(\vec{u}\) and \(\vec{v}\) defined below:
\[\vec{u} = \begin{bmatrix} 1 \\ -3 \\ 8 \end{bmatrix} \qquad \vec{v} = \begin{bmatrix} 3 \\ 0 \\ -1 \end{bmatrix}\]Determine the values of the following quantities.
(a) \(\lVert \vec u \rVert\)
(b) \(\vec u \cdot \vec v\)
(c) \(\vec v^T \vec u\)
Question 1.2
Suppose that on your way home from discussion section on North Campus, you drive 2 miles east and 7 miles north. (For the purposes of this question, assume that North Campus can be represented by just a single point.)
(a) How far do you live from North Campus, in miles?
(b) Suppose we draw a horizontal line passing through North Campus, and a line passing through North Campus and your home. Determine the angle between the two lines in degrees. (You’ll need to use a calculator – Google works just fine!)
Question 1.3
Suppose \(\vec{1} \in \mathbb{R}^n\) is a vector containing the value 1 for each element, i.e. \(\vec{1} = \begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}\). For any other vector \(\vec{b} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}\), what is the value of \(\vec{1} \cdot \vec{b}\)?
The dot product, angles, and orthogonality
Practice Questions 2.1-2.3
Question 2.1
Consider the vectors \(\vec{u}\) and \(\vec{v}\) defined below:
\[\vec{u} = \begin{bmatrix} 1 \\ -3 \\ 8 \end{bmatrix} \qquad \vec{v} = \begin{bmatrix} 3 \\ 0 \\ -1 \end{bmatrix}\]Determine the angle between \(\vec u\) and \(\vec v\) in the form of \(\cos^{-1} ( \cdot )\). Hint: You’ll know you did this correctly if you find that, when converted to degrees, the angle between \(\vec u\) and \(\vec v\) is approximately \(101º\).
Question 2.2
Note: In addition to reviewing the concepts in the video, this question is also designed to refresh your understanding of limits from Calculus 1.
Consider the vectors \(\vec{x}\) and \(\vec{y}_c\) defined below:
\[\vec{x} = \begin{bmatrix} -4 \\ 3 \end{bmatrix} \qquad \vec{y}_c = \begin{bmatrix} 3 \\ c \end{bmatrix}\]Here, \(c\) is an unknown real number. For example:
\[\vec{y}_2 = \begin{bmatrix} 3 \\ 2 \end{bmatrix}\](a) On a piece of paper (or on a tablet), draw \(\vec x\), \(\vec y_0\), \(\vec y_{20}\), and \(\vec y_{-30}\).
(b) Define \(\theta_c\) to be the angle between \(\vec{x}\) and \(\vec{y}_c\). Using properties of limits, show that:
\[\lim_{c \rightarrow \infty} \theta_c = \cos^{-1} \left( \frac{3}{5} \right)\](c) Is \(\cos^{-1} \left( \frac{3}{5} \right)\) the largest possible value of \(\theta_c\), or the smallest possible value?
- If you believe this is the largest possible value of \(\theta_c\), determine the smallest possible value of \(\theta_c\).
- If you believe this is the smallest possible value of \(\theta_c\), determine the largest possible value of \(\theta_c\).
(d) \(\cos^{-1} \left( \frac{3}{5} \right)\) – which, recall, is \(\displaystyle \lim_{c \rightarrow \infty} \theta_c\) – is also equal to the angle between \(\vec{x}\) and a particular unit vector, \(\vec u\). (A unit vector \(\vec u\) is a vector such that \(\lVert \vec u \rVert\) = 1.) What is the vector \(\vec u\)?
Question 2.3
As we saw in the first two videos, the dot product \(\vec u \cdot \vec v\) of two vectors \(\vec u, \vec v \in \mathbb{R}^n\):
- returns a scalar – that is, a single number.
- is valid for any \(n \geq 1\), as long as both \(\vec u\) and \(\vec v\) have the same number of components.
- measures how similar \(\vec u\) and \(\vec v\) are.
The cross product \(\vec u \times \vec v\) of two vectors is only defined when both vectors are in \(\mathbb{R}^3\). If \(\vec{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}\) and \(\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}\), then:
\[\vec u \times \vec v = \begin{bmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{bmatrix}\]Note that the cross product of two vectors in \(\mathbb{R}^3\) is another vector in \(\mathbb{R}^3\), rather than a scalar! In particular, the cross product \(\vec u \times \vec v\) is defined to be a vector orthogonal to both \(\vec u\) and \(\vec v\), with a length of \(\lVert \vec u \rVert \lVert \vec v \rVert \sin \theta\), pointing in the direction determined by the right hand rule:
(a) Prove that \(\vec u \times \vec v\) is orthogonal to both \(\vec u\) and \(\vec v\).
(b) What is the vector \(\vec u \times \vec u\)?
(c) Once again, consider the vectors \(\vec{u}\) and \(\vec{v}\) defined below:
\[\vec{u} = \begin{bmatrix} 1 \\ -3 \\ 8 \end{bmatrix} \qquad \vec{v} = \begin{bmatrix} 3 \\ 0 \\ -1 \end{bmatrix}\]There are infinitely many vectors that are orthogonal to both \(\vec u\) and \(\vec v\), but they all point in the same direction. Determine the vector \(\vec w = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}\) that is orthogonal to both \(\vec u\) and \(\vec v\) such that \(w_1 + w_2 + w_3 = 1\).