The dot product

The dot product or scalar product defined geometrically, the angle between two vectors, perpendicular vectors, the algebraic laws for the dot product, and using it to prove geometric results.

The geometric definition

Dot product

$$\underset{\sim}{u} \cdot \underset{\sim}{v} = |\underset{\sim}{u}|\,|\underset{\sim}{v}|\cos\theta$$

Here $\theta$ is the angle between the vectors when they are placed tail to tail, with $0^\circ \le \theta \le 180^\circ$. The result is a scalar, not a vector.

$\underset{\sim}{u} \cdot \underset{\sim}{v} \gt 0$ when $\theta$ is acute, and $\lt 0$ when $\theta$ is obtuse.
Non-zero vectors are perpendicular exactly when $\underset{\sim}{u} \cdot \underset{\sim}{v} = 0$.
$\underset{\sim}{u} \cdot \underset{\sim}{u} = |\underset{\sim}{u}|^2$.

Example 1●

$|\underset{\sim}{u}| = 3$, $|\underset{\sim}{v}| = 4$ and the angle between them is $60^\circ$. Find $\underset{\sim}{u} \cdot \underset{\sim}{v}$.

$3 \times 4 \times \cos 60^\circ = 6$.

The laws

Commutative	$\underset{\sim}{u} \cdot \underset{\sim}{v} = \underset{\sim}{v} \cdot \underset{\sim}{u}$
Distributive	$\underset{\sim}{u} \cdot (\underset{\sim}{v} + \underset{\sim}{w}) = \underset{\sim}{u} \cdot \underset{\sim}{v} + \underset{\sim}{u} \cdot \underset{\sim}{w}$
Scalars	$(\lambda\underset{\sim}{u}) \cdot \underset{\sim}{v} = \lambda(\underset{\sim}{u} \cdot \underset{\sim}{v})$

These let you expand dot products like ordinary brackets.

Example 2: the cosine rule●●

Expand $|\underset{\sim}{a} - \underset{\sim}{b}|^2$.

$(\underset{\sim}{a} - \underset{\sim}{b}) \cdot (\underset{\sim}{a} - \underset{\sim}{b}) = |\underset{\sim}{a}|^2 - 2\,\underset{\sim}{a} \cdot \underset{\sim}{b} + |\underset{\sim}{b}|^2$.

With $\underset{\sim}{a} \cdot \underset{\sim}{b} = |\underset{\sim}{a}|\,|\underset{\sim}{b}|\cos\theta$, this is the cosine rule $c^2 = a^2 + b^2 - 2ab\cos\theta$.

Example 3: a geometric proof●●●

Prove that the diagonals of a rhombus are perpendicular.

Let the rhombus be $OABC$ with $\overrightarrow{OA} = \underset{\sim}{a}$, $\overrightarrow{OC} = \underset{\sim}{c}$ and $|\underset{\sim}{a}| = |\underset{\sim}{c}|$.

The diagonals are $\overrightarrow{OB} = \underset{\sim}{a} + \underset{\sim}{c}$ and $\overrightarrow{AC} = \underset{\sim}{c} - \underset{\sim}{a}$.

$(\underset{\sim}{a} + \underset{\sim}{c}) \cdot (\underset{\sim}{c} - \underset{\sim}{a}) = |\underset{\sim}{c}|^2 - |\underset{\sim}{a}|^2 = 0$, so the diagonals are perpendicular.

Flashcards

$underset{sim}{u} cdot underset{sim}{v}$ (geometric)

$|\underset{\sim}{u}|\,|\underset{\sim}{v}|\cos\theta$

Perpendicular test

$\underset{\sim}{u} \cdot \underset{\sim}{v} = 0$

$underset{sim}{u} cdot underset{sim}{u}$

$|\underset{\sim}{u}|^2$

$(underset{sim}{a} + underset{sim}{b}) cdot (underset{sim}{a} - underset{sim}{b})$

$|\underset{\sim}{a}|^2 - |\underset{\sim}{b}|^2$

Practice

$|\underset{\sim}{u}| = 3$, $|\underset{\sim}{v}| = 4$ and the angle between them is $60^\circ$. Find $\underset{\sim}{u} \cdot \underset{\sim}{v}$.
$|\underset{\sim}{u}| = 5$, $|\underset{\sim}{v}| = 4$ and the angle is $120^\circ$. Find $\underset{\sim}{u} \cdot \underset{\sim}{v}$.
$\underset{\sim}{u}$ and $\underset{\sim}{v}$ are perpendicular. Find $\underset{\sim}{u} \cdot \underset{\sim}{v}$.
$|\underset{\sim}{u}| = 7$. Find $\underset{\sim}{u} \cdot \underset{\sim}{u}$.
$|\underset{\sim}{u}| = 2$, $|\underset{\sim}{v}| = 5$ and $\underset{\sim}{u} \cdot \underset{\sim}{v} = 5$. Find the angle between them, in degrees.
$|\underset{\sim}{a}| = 4$ and $|\underset{\sim}{b}| = 3$. Find $(\underset{\sim}{a} + \underset{\sim}{b}) \cdot (\underset{\sim}{a} - \underset{\sim}{b})$.
$\underset{\sim}{u} \cdot \underset{\sim}{v} \lt 0$. The angle between the vectors is:
The dot product of two vectors is:
$|\underset{\sim}{a} + \underset{\sim}{b}|^2 = $