Dot Product and its Properties

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 2 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.

We will prove Property  (b). The remaining properties are left as exercises.

\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*}

We will illustrate Property (f) with an example.

If we take the square root of both sides of the equation in Property (f), we get an alternative way to think of the length of a vector.

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 (f) of Theorem 4.
Prove Properties (a), (c), (d), (e) and (f) of Theorem 4.
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.
\(\vec {u}=\begin{bmatrix}1\\\frac {1}{2}\end{bmatrix}\), \(\vec {v}=\begin{bmatrix}-2\\4\end{bmatrix}\) \(\vec {u}=\begin{bmatrix}-1\\\frac {1}{2}\end{bmatrix}\), \(\vec {v}=\begin{bmatrix}-2\\4\end{bmatrix}\) \(\vec {u}=\begin{bmatrix}1\\\frac {1}{2}\end{bmatrix}\), \(\vec {v}=\begin{bmatrix}1\\-2\end{bmatrix}\) \(\vec {u}=\begin{bmatrix}-1\\-\frac {1}{2}\end{bmatrix}\), \(\vec {v}=\begin{bmatrix}-2\\4\end{bmatrix}\)
Compute \(\vec {u}\dotp \vec {v}\) for each pair. What do you observe?
(a)
Vector \(\vec {u}\) parallel to 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 \(\vec {v}\) perpendicular to \(y=mx\).
What do you know about the slopes of perpendicular lines?
(b)
Find \(\vec {u}\dotp \vec {v}\).
(c)
What is true about the dot product of perpendicular vectors?