Skip to main content

Calculus 3 (Math 221) Notes and Study Guide

Brenton LeMesurier

Section 2.3 The Dot Product (a.k.a. Scalar Product)

Revised on March 14.

References.

Introduction.

There are two useful notions of a product of two vectors; in this section we meet the first of them:

Definition 2.3.1.

The dot product of two vectors in \(\mathbb{R}^3\) is
\begin{equation} \vecu \cdot \vecv = \vector{u_1, u_2, u_3}\cdot \vector{v_1, v_2, v_3}= u_1 v_1 + u_2 v_2 + u_3 v_3\tag{2.3.1} \end{equation}
with the obvious 2D version. This is also known as the scalar product (because the value is a scalar, not another vector) or the inner product.

Geometric Characterization.

Much of the importance of the dot product comes from its geometric properties; in fact the formula can be derived by requiring that
If \(\theta\) is the angle between the two vectors \(\vecu\) and \(\vecv\) then
\begin{equation} \vecu \cdot \vecv = \|\vecu\| \|\vecv\| \cos \theta.\tag{2.3.2} \end{equation}
along with some basic properties of a product:
  • \(\vecu \cdot (\vecv + \vecw)= \vecu \cdot \vecv + \vecu \cdot \vecw\text{:}\) (Distributive property)
  • \((a \vecu) \cdot \vecv = \vecu \cdot (a \vecv) = a (\vecu \cdot \vecv)\) (Associative property)
(In fact, (2.3.2) could be used as an equivalent definition.)
The main step is working out what (2.3.2) implies for the basic vectors \(\veci\text{,}\) \(\vecj\) and \(\veck\text{.}\)
Firstly,
\begin{equation*} \veci \cdot \veci = \vecj \cdot \vecj = \veck \cdot \veck = 1 \end{equation*}
because the angle is \(\theta = 0\) and the lengths are all 1.
Next,
\begin{equation*} \veci \cdot \vecj = \vecj \cdot \veci = \veci \cdot \veck = \veck \cdot \veci = \vecj \cdot \veck = \veck \cdot \veci = 0 \end{equation*}
because now all the angles are \(\pi/2\) so with zero cosine. Note that commutativity is not assumed (and will fail for the cross product introduced in the next section); it is instead verified as a consequence of the definition.
Finally
\begin{equation*} \begin{split} \vecu \cdot \vecv \amp= (u_1 \veci + u_2 \vecj + u_3 \veck) \cdot (v_1 \veci + v_2 \vecj + v_3 \veck) \\ \amp= (u_1 \veci) \cdot (v_1 \veci) + (u_2 \vecj) \cdot (v_2 \vecj) + (u_3 \veck) \cdot (v_3 \veck) + (u_1 \veci) \cdot (v_2 \vecj) + \cdots \\ \amp= u_1 v_1 \veci \cdot \veci + u_2 v_2 \vecj \cdot \vecj + u_3 v_3 \veck \cdot \veck + u_1 v_2 \veci \cdot \vecj + \cdots \\ \amp= u_1 v_1 + u_2 v_2 + u_3 v_3 \end{split} \end{equation*}
as all the remaining terms involve zero dot products.

Properties.

From the formula in definition Definition 2.3.1 on can derive some familiar properties (including the one assumed to show the connection with the geometric formula (2.3.2)):
  1. \(\displaystyle \vecu \cdot \vecv = \vecv \cdot \vecu \qquad \text{Commutativity}\)
  2. \(\displaystyle \vecu \cdot (\vecv + \vecw)= \vecu \cdot \vecv + \vecu \cdot \vecw \qquad \text{Distributivity}\)
  3. \(\displaystyle (a \vecu) \cdot \vecv = \vecu \cdot (a \vecv) = a (\vecu \cdot \vecv) \qquad \text{Associativity with scalar multiplication}\)
  4. \(\displaystyle \vec{u} \cdot \vec{0} = 0\)
  5. \(\displaystyle \vecu \cdot \vecu = \|\vecu\|^2\)

Direction Angles and Direction Cosines.

The direction angles of a non-zero vector \(\vecu\) are the angles that it makes with the three coordinate axes, typically called \(\alpha\text{,}\) \(\beta\) and \(\gamma\text{.}\) That is, the angles that the vector makes with the three standard basic vectors \(\veci\text{,}\) \(\vecj\) and \(\hat{k}\text{.}\) The cosines of these are the direction cosines.
\begin{equation*} \cos \alpha = \frac{\vecu \cdot \veci}{|\vecu| |\veci|}, = \frac{u_1}{|\vecu|}, %\text{ (so } \alpha =\arccos\left(\frac{u_1}{|\vecu|}\right) ), \qquad \cos \beta = \frac{u_2}{|\vecu|}, \qquad \cos \gamma = \frac{u_3}{|\vecu|} \end{equation*}
Since the angles are always in \([0, \pi]\text{,}\) they are unambiguously determined by their cosines; thus is usually enough to know the direction cosines. It can be checked that
\begin{equation*} \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma=1, \quad \vecu = |\vecu| \vector{\cos \alpha, \cos \beta, \cos \gamma} \end{equation*}
so the vector \(\hat{u}=\vector{\cos \alpha, \cos \beta, \cos \gamma}\) of direction cosines is the unit vector in the direction of \(\vecu\text{.}\)

Projections.

The vector projection (or just projection) of \(\vecv\) onto \(\vecu\) is the vector that is parallel to \(\vecu\) (a scalar multiple of \(\vecu\)) and with the difference between it and \(\vecv\) being perpendicular to \(\vecu\text{.}\) Calling this vector \(\vec{p}\text{,}\) these conditions are that \(\vec{p} = c \vecu\) and \((\vecv-\vec{p}) \cdot \vecu=0\text{.}\)
These conditions have a unique solution, denoted \(\mbox{proj}_{\vecu} \vecv\text{:}\)
\begin{equation*} \text{proj}_{\vecu} \vecv = \frac{\vecu \cdot \vecv}{|\vecu|^2} \vecu \end{equation*}
The scalar projection of \(\vecv\) onto \(\vecu\) is the "signed magnitude" of this: the magnitude with sign plus or minus according to whether the projection goes in the same or opposite direction as \(\vecu\text{.}\)
This is also called the component of \(\vecv\) in direction \(\vecu\), and denoted
\begin{equation*} \text{comp}_{\vecu} \vecv = \pm |\text{proj}_{\vecu} \vecv| = \frac{\vecu \cdot \vecv}{|\vecu|} = |\vecv|\cos \theta, \quad \theta\text{ the angle between }\vecu\text{ and }\vecv. \end{equation*}

Study Guide.

Study Section 2.3 of Calculus Volume 3
 6 
openstax.org/books/calculus-volume-3/pages/2-3-the-dot-product
; in particular:
  • The Definitions and Theorems.
  • Examples 2.21–2.25, 2.27 and 2.28, and the Checkpoints following each. (Examples 2.29 and 2.30 also show connections to Physics.)
  • One or several exercises from each of the following ranges: One or several exercises from each of the ranges 123–126, 131–134, 135–140, 141–144, 147–148, 149–150, 161–164, 167–170, 171–172.
PrevTopNext