Learning Outcomes
- Use the dot product to determine norms, distances, and angles.
Section 6.1 Dot Products (ON1)
Activity 6.1.1.
Consider the binary operation ’’\(\circledast\)’’ defined on vectors \(\vec v\text{,}\) \(\vec w\) in \(\IR^n\) by
\begin{equation*}
\vec v \circledast \vec w = v_1 w_1 + v_2 w_2 + v_3 w_3 + \cdots + v_n w_n.
\end{equation*}
(a)
Let \(\vec v =\left[\begin{array}{c} 1 \\ -1 \\ 0 \\ 2 \\ 3 \end{array}\right]\) and \(\vec w=\left[\begin{array}{c} 5 \\ 12 \\ -1 \\ 1 \\ 2 \end{array}\right]\text{.}\) What is \(\vec v \circledast \vec w\text{?}\)
- \(\displaystyle 25\)
- \(\displaystyle \left[\begin{array}{c} 5 \\ -12 \\ 0 \\ 2 \\ 6 \end{array}\right]\)
- \(\displaystyle 1\)
- \(\displaystyle \left[\begin{array}{c} 6 \\ 11 \\ -1 \\ 3 \\ 5 \end{array}\right]\)
(b)
Let \(\vec v=\left[\begin{array}{c} 48 \\ 55 \end{array}\right]\text{.}\)
- Graph \(\vec v\) and use the Pythagorean Theorem to determine the length of \(\vec v\text{.}\)
- What is \(\vec v \circledast \vec v\text{?}\)
(c)
Let \(\vec v=\left[\begin{array}{c} v_1 \\ v_2 \end{array}\right]\text{.}\)
- Graph \(\vec v\) and use the Pythagorean Theorem to determine the length of \(\vec v\text{.}\)
- What is \(\vec v \circledast \vec v\text{?}\)
- For a vector \(\vec v\) in \(\mathbb{R}^2\text{,}\) how is the length of \(\vec v\) related to \(\vec v \circledast \vec v\text{?}\)
Subsection 6.1.1 The Dot Product
The dot product is a binary operation on vectors that helps us measure the length of vectors and the angle formed by a pair of vectors.
Definition 6.1.2.
Given two \(n\)-dimensional vectors \(\vec v\) and \(\vec w\text{,}\) the dot product \(\vec v \cdot \vec w\) is defined by
\begin{equation*}
\vec v \cdot \vec w = \left[\begin{array}{c} v_1 \\ v_2 \\ v_3 \\ \vdots \\ v_n \end{array}\right] \cdot \left[\begin{array}{c} w_1 \\ w_2 \\ w_3 \\ \vdots \\ w_n \end{array}\right]
= v_1 w_1 + v_2 w_2 + v_3 w_3 + \cdots + v_n w_n
\end{equation*}
The dot product combines two vectors and creates a scalar that gives us geometric information about the input vectors. If both vectors are the same, then \(\vec{v} \cdot \vec{v}\) gives us the square of the length of \(\vec{v}\text{.}\) The length of a vector \(\vec v\) in \(\IR^n\text{,}\) denoted \(\lvert \vec v \rvert\text{,}\) is defined as
\begin{equation*}
\lvert \vec v \rvert = \sqrt{\vec v \cdot \vec v}=\sqrt{v_1^2 + v_2^2 + v_3 ^2 + \cdots + v_n^2}
\end{equation*}
Vectors of length \(1\) are called unit vectors.
Activity 6.1.3.
Consider each of the following properties of the dot product. Label each property as valid if the property holds for Euclidean vectors \(\vec u\text{,}\) \(\vec v\) and \(\vec w\) from \(\IR^n\text{,}\) and scalars \(a,b \in \IR\text{,}\) and invalid if it does not.
- \(\left(\vec u \cdot \vec v\right) \cdot \vec w=\vec u \cdot \left(\vec v \cdot \vec w\right)\text{.}\)
- \(\left(a \vec v\right) \cdot \vec w=a \left(\vec v \cdot \vec w\right)\text{.}\)
- \(\vec w\cdot \vec v=\vec v \cdot \vec w\text{.}\)
- \(\left(a+b\right) \left(\vec v \cdot \vec w\right)= \left(a \vec v\right) \cdot \left(b \vec w\right)\text{.}\)
- \(\left(a \vec u+b \vec v\right) \cdot \vec w= \left(a \vec u\right) \cdot \vec w + \left(b \vec v\right) \cdot \vec w\text{.}\)
Observation 6.1.4.
Like arithmetic of real numbers, the dot product on vectors satisfies some familiar properties. Let \(\vec u\text{,}\) \(\vec v\) and \(\vec w\) be vectors from \(\IR^n\text{,}\) and let \(a,b \in \IR\) be scalars.
- \(\vec u \cdot \vec v = \vec v \cdot \vec u\text{.}\)
- \(\left( a\vec u\right) \cdot \vec v = a\left(\vec u \cdot \vec v\right)\text{.}\)
- \(\left(a\vec u + b \vec v\right)\cdot \vec w =a \vec u \cdot \vec w + b \vec v \cdot \vec w\text{.}\)
Activity 6.1.5.
Given the linear transformation \(S:\IR^2 \to \IR^2\) whose standard matrix is \(\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right]\) and vector \(\vec v = \left[\begin{array}{c} 3 \\ 4 \end{array}\right]\text{,}\)
(a)
Graph \(\vec v\) and \(S( \vec v )\text{.}\)
(b)
For an unspecified vector \(\vec w = \left[\begin{array}{c} w_1 \\ w_2 \end{array}\right]\text{,}\) describe the relationship between \(\vec w\) and \(S( \vec w )\text{.}\)
Activity 6.1.6.
Consider \(\vec v = \left[\begin{array}{c} 3 \\ 4 \end{array}\right]\text{.}\)
(a)
What vector \(\vec w = \left[\begin{array}{c} ? \\ ? \end{array}\right]\) is the result of rotating \(\vec v\) by \(90^{\circ}\) counter-clockwise?
(b)
Find the dot product \(\vec v \cdot \vec w\text{.}\)
(c)
For an arbitrary vector \(\vec x = \left[\begin{array}{c} x_1 \\ x_2 \end{array}\right]\text{,}\) what vector \(\vec y = \left[\begin{array}{c} ? \\ ? \end{array}\right]\) is the result of rotating \(\vec x\) by \(90^{\circ}\) counter-clockwise?
(d)
Find the dot product \(\vec x \cdot \vec y\text{.}\)
(e)
Suppose two vectors are perpendicular. What can you say about their dot product?
Definition 6.1.7.
Two vectors \(\vec u\) and \(\vec v\) in \(\IR^n\) are orthogonal provided \(\vec u \cdot \vec v = 0\text{.}\)
Definition 6.1.8.
Given two vectors \(\vec u\) and \(\vec v\) in \(\IR^n\text{,}\) the distance between the vectors, denoted \(d(\vec u,\vec v)\) is given by
\begin{equation*}
d(\vec u,\vec v)=\lvert \vec u-\vec v\rvert.
\end{equation*}
Activity 6.1.9.
Consider the vectors \(\vec u=\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(\vec v=\left[\begin{array}{c} 1 \\ 3 \end{array}\right].\)
(a)
Draw the triangle created by the two vectors in \(\IR^2.\) What vector represents the third side of the triangle? Is the answer unique?
(b)
Find the length of each side of the triangle.
(c)
Calculate the distance between \(\vec u\) and \(\vec v\text{.}\)
(d)
Does the Pythagorean Theorem hold for this triangle?
(e)
There exists a pair of orthogonal vectors in the triangle from part (a).
- True
- False
Need to create a segue to the fact that you can find the angle between vectors using the Law of Cosines as a starting place.
Definition 6.1.10.
Given two vectors \(\vec u\) and \(\vec v\) in \(\IR^n\text{,}\) such that \(\vec u\) and \(\vec v\) are not parallel, let \(\theta\) be the angle between the two vectors, then
\begin{equation*}
\cos \theta = \frac{\vec u \cdot \vec v }{|\vec u||\vec v|}.
\end{equation*}
By the Law of Cosine:
\begin{equation*}
|\vec u- \vec v|^2 = |\vec u|^2+|\vec v|^2- 2 |\vec u||\vec v| \cos \theta.
\end{equation*}
Using dot product of vector \(\vec u-\vec v\) with itself:
\begin{align*}
|\vec u- \vec v|^2 \amp = (\vec u- \vec v) \cdot (\vec u- \vec v) \amp \\
\amp = \vec u \cdot \vec u - 2 ( \vec u \cdot \vec v) + \vec v \cdot \vec v \amp \\
\amp = |\vec u|^2 - 2 ( \vec u.\vec v) + |\vec v|^2 \amp
\end{align*}
Hence from above, we have:
\begin{equation*}
\vec u \cdot \vec v = |\vec u||\vec v| \cos \theta .
\end{equation*}
One of these activity should be deleted from here.
Activity 6.1.11.
Suppose that \(\vec u =\left[\begin{array}{c} 4 \\ -1 \\ 0 \end{array}\right]\) and \(\vec v = \left[\begin{array}{c} 2 \\ 3 \\ 1 \end{array}\right]\text{.}\)
(a)
Find the length of \(\vec u\) and the length of \(\vec v\text{.}\)
(b)
Describe all vectors \(\vec w\) that are orthogonal to \(\vec u\text{.}\)
(c)
Find the angle between \(\vec u\) and \(\vec v\text{.}\)
Activity 6.1.12.
Consider two vectors \(\vec u =\left[\begin{array}{c} 1 \\ 3 \\ 4 \\ -4 \end{array}\right]\) and \(\vec v = \left[\begin{array}{c} -1 \\ -3 \\ -4 \\ 4 \end{array}\right]\text{.}\)
(a)
Use dot product to determine \(|\vec u| \) and \(|\vec v|\text{.}\)
(b)
Using dot product, find the distance between \(\vec u\) and \(\vec v \text{.}\)
(c)
Find the angle between \(\vec u\) and \(\vec v\text{.}\)
Subsection 6.1.2 Videos
Exercises 6.1.3 Exercises
Exercises available at checkit.clontz.org.
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