The dot product with components

The component formula for the dot product in two and three dimensions, its proof from the distributive law, finding angles between vectors, testing for perpendicular vectors, and finding unknown components.

The component formula

Dot product with components

$$\begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix} \cdot \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix} = x_1x_2 + y_1y_2 + z_1z_2$$

In two dimensions, leave out the $z$ terms. The formula follows from the distributive law, because $\underset{\sim}{i} \cdot \underset{\sim}{i} = \underset{\sim}{j} \cdot \underset{\sim}{j} = \underset{\sim}{k} \cdot \underset{\sim}{k} = 1$, while the dot product of any two different ones is $0$.

Example 1●●

Find the angle between $\underset{\sim}{u} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ and $\underset{\sim}{v} = \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$.

$\underset{\sim}{u} \cdot \underset{\sim}{v} = 2 - 2 + 4 = 4$, and $|\underset{\sim}{u}| = |\underset{\sim}{v}| = 3$.

$\cos\theta = \tfrac49$, so $\theta \approx 63.6^\circ$.

Example 2●

Show that $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$ are perpendicular.

$3 \times 4 + 4 \times (-3) = 0$, so they are perpendicular.

Example 3●●

Find $k$ so that $\begin{pmatrix} 2 \\ k \\ -1 \end{pmatrix}$ is perpendicular to $\begin{pmatrix} 3 \\ 2 \\ 4 \end{pmatrix}$.

$6 + 2k - 4 = 0$, so $k = -1$.

Example 4: an angle in a triangle●●●

Find $\angle ABC$ for $A(1, 0)$, $B(3, 2)$ and $C(5, 0)$.

Use the vectors from the vertex $B$: $\overrightarrow{BA} = \begin{pmatrix} -2 \\ -2 \end{pmatrix}$ and $\overrightarrow{BC} = \begin{pmatrix} 2 \\ -2 \end{pmatrix}$.

$\overrightarrow{BA} \cdot \overrightarrow{BC} = -4 + 4 = 0$, so $\angle ABC = 90^\circ$.

Flashcards

$underset{sim}{u} cdot underset{sim}{v}$ with components

$x_1x_2 + y_1y_2 + z_1z_2$

$cos heta$ between two vectors

$\dfrac{\underset{\sim}{u} \cdot \underset{\sim}{v}}{|\underset{\sim}{u}|\,|\underset{\sim}{v}|}$

$underset{sim}{i} cdot underset{sim}{j}$

$0$

Practice

Find $\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$.
Find the angle between those vectors, in degrees correct to one decimal place.
Find $\begin{pmatrix} 3 \\ 4 \end{pmatrix} \cdot \begin{pmatrix} 4 \\ -3 \end{pmatrix}$.
Find $k$ so that $\begin{pmatrix} 2 \\ k \\ -1 \end{pmatrix}$ is perpendicular to $\begin{pmatrix} 3 \\ 2 \\ 4 \end{pmatrix}$.
Find $\angle ABC$, in degrees, for $A(1, 0)$, $B(3, 2)$ and $C(5, 0)$.
Find the angle between $\underset{\sim}{i}$ and $\underset{\sim}{i} + \underset{\sim}{j}$, in degrees.
Find $\underset{\sim}{k} \cdot \underset{\sim}{k}$.
To find the angle at vertex $B$ of triangle $ABC$, use:
$\underset{\sim}{u} \cdot \underset{\sim}{v} = |\underset{\sim}{u}|\,|\underset{\sim}{v}|$. The vectors are: