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  2. 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 6 with an example.

If we take the square root of both sides of the equation in Property 6, 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 6 of Theorem 4.
Prove Properties 1, 3, 4, 5 and 6 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?
1.
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?
2.
Find \(\vec {u}\dotp \vec {v}\).
3.
What is true about the dot product of perpendicular vectors?