Dot Product and its Properties
Let and be vectors in . The dot product of and , denoted by , 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*}
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, . This and other properties of the dot product are stated below.
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*}
We will illustrate Property item:norm with an example.
If we take the square root of both sides of the equation in Property item:norm, we get an alternative way to think of the length of a vector.
Length of a Vector Let be a vector in , then the length, or the magnitude, of is given
by \begin{equation*} \label{eq:norm_dotp} \norm{\vec{v}}=\sqrt{\vec{v} \dotp \vec{v}} \end{equation*}
Practice Problems
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. Compute for each pair. What do you observe?
Problems prob:perpvectors2a-prob:perpvectors2c
For each problem below
- (a)
- Find the value of that will make vectors and perpendicular. (Hint: Think of the -component as the “run” and the -component as the “rise”, then use what you know about slopes of perpendicular lines.)
- (b)
- Find .
- (a)
- Vector that lies on the line , has the form . Assuming that , find the general form for a vector that lies on a line perpendicular to .
- (b)
- Find .
- (c)
- Formulate a conjecture about the dot product of perpendicular vectors.