VEC-0050: Dot Product and its Properties

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We define the dot product and prove its algebraic properties.

VEC-0050: Dot Product and its Properties

Let $\vec {u}$ and $\vec {v}$ be vectors in $\RR ^n$ . The dot product of $\vec {u}$ and $\vec {v}$ , denoted by $\vec {u}\dotp \vec {v}$ , is given by $\begin{align*} \vec{u}\dotp\vec{v}=\begin{bmatrix}u_1\\u_2\\\vdots\\u_n\end{bmatrix}\dotp\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}=u_1v_1+u_2v_2+\ldots+u_nv_n. \end{align*}$

Find $\vec {u}\dotp \vec {v}$ if $\vec {u}=\begin {bmatrix}-2\\0\\1\end {bmatrix}$ and $\vec {v}=\begin {bmatrix}3\\2\\-4\end {bmatrix}$ .

$\vec {u}\dotp \vec {v}=\begin {bmatrix}-2\\0\\1\end {bmatrix}\dotp \begin {bmatrix}3\\2\\-4\end {bmatrix}=(-2)(3)+(0)(2)+(1)(-4)=-6-4=-10$

Note that the dot product of two vectors is a scalar. For this reason, the dot product is sometimes called a scalar product.

Properties of the Dot Product

A quick examination of Example ex:dotex will convince you that the dot product is commutative. In other words, $\vec {u}\dotp \vec {v}=\vec {v}\dotp \vec {u}$ . This and other properties of the dot product are stated below.

The following properties hold for vectors $\vec {u}$ , $\vec {v}$ and $\vec {w}$ in $\RR ^n$ and scalar $k$ .

(a): $\vec {u}\dotp \vec {v}=\vec {v}\dotp \vec {u}$
(b): $(\vec {u}+\vec {v})\dotp \vec {w}=\vec {u}\dotp \vec {w}+\vec {v}\dotp \vec {w}$
(c): $\vec {u}\dotp (\vec {v}+\vec {w})=\vec {u}\dotp \vec {v}+\vec {u}\dotp \vec {w}$
(d): $(k\vec {u})\dotp \vec {v}=k(\vec {u}\dotp \vec {v})=\vec {u}\dotp (k\vec {v})$
(e): $\vec {u}\dotp \vec {u}\geq 0$ , and $\vec {u}\dotp \vec {u}=0$ if and only if $\vec {u}={\bf 0}$ .
(f): $\norm {\vec {u}}^2=\vec {u}\dotp \vec {u}$

We will prove Property item:distributive. The remaining properties are left as exercises.

Proof of Property item:distributive:: $\begin{align*} \left(\vec{u}+\vec{v}\right)\dotp \vec{w}&=\left(\begin{bmatrix} u_1\\ u_2\\ \vdots\\ u_n \end{bmatrix}+\begin{bmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{bmatrix}\right)\dotp \begin{bmatrix}w_1\\w_2\\\vdots\\w_n\end{bmatrix}=\begin{bmatrix} u_1+v_1\\ u_2+v_2\\ \vdots\\ u_n+v_n \end{bmatrix}\dotp \begin{bmatrix}w_1\\w_2\\\vdots\\w_n\end{bmatrix}\\ &=(u_1+v_1)w_1+ (u_2+v_2)w_2+ \ldots+ (u_n+v_n)w_n\\ &=u_1w_1+v_1w_1+ u_2w_2+v_2w_2+ \ldots+ u_nw_n+v_nw_n\\ &=(u_1w_1+ u_2w_2\ldots+u_nw_n)+(v_1w_1+v_2w_2+ \ldots +v_nw_n)\\ &=\begin{bmatrix} u_1\\ u_2\\ \vdots\\ u_n \end{bmatrix}\dotp\begin{bmatrix}w_1\\w_2\\\vdots\\w_n\end{bmatrix}+\begin{bmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{bmatrix}\dotp \begin{bmatrix}w_1\\w_2\\\vdots\\w_n\end{bmatrix} =\vec{u}\dotp\vec{w}+\vec{v}\dotp\vec{w} \end{align*}$ $\blacksquare$

We will illustrate Property item:norm with an example.

Let $\vec {u}=\begin {bmatrix}-2\\3\end {bmatrix}$ . Use $\vec {u}$ to illustrate Property item:norm of Theorem th:dotproductproperties.

$\norm {\vec {u}}^2=(-2)^2+3^2=(-2)(-2)+(3)(3)=\vec {u}\dotp \vec {u}$

Practice Problems

Find the dot product of $\vec {u}$ and $\vec {v}$ if $\vec {u}=\begin {bmatrix}-1\\-2\\5\\4\end {bmatrix},\quad \vec {v}=\begin {bmatrix}2\\-2\\-3\\1\end {bmatrix}$ Answer: $\vec {u} \dotp \vec {v} = \answer {-9}$

Find the dot product of $\vec {u}$ and $\vec {v}$ if $\vec {u}=\begin {bmatrix}1\\1/2\end {bmatrix},\quad \vec {v}=\begin {bmatrix}-2\\4\end {bmatrix}$ Answer:

$\vec {u} \dotp \vec {v} = \answer {0}$

Use vector $\vec {u}=\begin {bmatrix}2\\5\\-7\end {bmatrix}$ to illustrate Property item:norm of Theorem th:dotproductproperties.

Prove Properties item:commutative, item:distributive-again, item:scalar, item:positive and item:norm of Theorem th:dotproductproperties.

From the given list of vector pairs, identify ALL pairs of vectors that lie on perpendicular lines.

You may want to draw a picture and think about what you know about slopes of perpendicular lines.

Compute $\vec {u}\dotp \vec {v}$ for each pair. What do you observe?

For each part below

(a): Find the value of $x$ that will make vectors $\vec {u}$ and $\vec {v}$ perpendicular.
Think of the $x$ -component as the “run” and the $y$ -component as the “rise”, then use what you know about slopes of perpendicular lines.
(b): Find $\vec {u}\dotp \vec {v}$ .

$\vec {u} = \begin {bmatrix}1\\2\end {bmatrix},\quad \vec {v}=\begin {bmatrix}2\\x\end {bmatrix}$

Answer: $x = \answer {-1}$ $\vec {u}\dotp \vec {v}=\answer {0}$

$\vec {u} = \begin {bmatrix}5\\2\end {bmatrix},\quad \vec {v}=\begin {bmatrix}x\\-4\end {bmatrix}$ Answer: $x = \answer {8/5}$ $\vec {u}\dotp \vec {v}=\answer {0}$

$\vec {u} = \begin {bmatrix} 4\\-3\end {bmatrix},\quad \vec {v} =\begin {bmatrix}6\\x\end {bmatrix}$ Answer: $x = \answer {8}$ $\vec {u}\dotp \vec {v}=\answer {0}$

(a): Vector $\vec {u}$ that lies on the line $y=mx$ , has the form $\vec {u}=k\begin {bmatrix}1\\m\end {bmatrix}$ . Assuming that $m\neq 0$ , find the general form for a vector $\vec {v}$ that lies on a line perpendicular to $y=mx$ .
What do you know about the slopes of perpendicular lines?
(b): Find $\vec {u}\dotp \vec {v}$ .
(c): Formulate a conjecture about the dot product of perpendicular vectors.

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