How the Dot Product Reveals the Angle Between Two Arrows

Vectors cannot be multiplied like ordinary numbers, but one useful product turns two vectors into a single number that measures how much they point the same way. This scalar, or dot, product gives angles between directions, tests for perpendicularity, and finds the work done by a force.

This lesson covers the definition and geometric meaning, properties and angles, and projections.

**The scalar product of $\vec{a}$ and $\vec{b}$ is $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta$, where $\theta$ is the angle between them, and in components it is $a_1{b}_1+a_2{b}_2+a_3{b}_3$; geometrically it is the length of one vector times the projection of the other on it.

Unit vector products:**

- $\hat{i}\cdot \hat{i}=\hat{j}\cdot \hat{j}=\hat{k}\cdot \hat{k}=1$
- $\hat{i}\cdot \hat{j}=\hat{j}\cdot \hat{k}=\hat{k}\cdot \hat{i}=0$

Worked example. For $\vec{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-2\hat{j}+4\hat{k}$:

$$\vec{a}\cdot \vec{b}=2-6-4=-8$$

The negative value shows that the angle between them is obtuse.

Physics link — work done. A force $\vec{F}=3\hat{i}+4\hat{j}$ newtons moving an object through $\vec{d}=5\hat{i}$ metres does work $\vec{F}\cdot \vec{d}=15$ joules — only the part of the force along the motion counts.

An everyday example. Pulling a loaded luggage trolley along a railway platform by a handle held at an angle moves it forward using only the part of your pull along the platform — the dot product measures that useful part.

The substance. The sign of the dot product tells the type of angle — positive for acute, zero for a right angle, negative for obtuse.

**The scalar product is commutative and distributive, $\vec{a}\cdot \vec{a}=|\vec{a}{|}^2$, non-zero vectors are perpendicular exactly when $\vec{a}\cdot \vec{b}=0$, and the angle between vectors is given by $\cos \theta =\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}$.

Properties:**

- $\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}$
- $\vec{a}\cdot (\vec{b}+\vec{c})=\vec{a}\cdot \vec{b}+\vec{a}\cdot \vec{c}$
- $|\vec{a}+\vec{b}{|}^2=|\vec{a}{|}^2+2\vec{a}\cdot \vec{b}+|\vec{b}{|}^2$

Worked example (angle). Find the angle between $\vec{a}=\hat{i}+\hat{j}$ and $\vec{b}=\hat{j}+\hat{k}$.

$$\cos \theta =\frac{0+1+0}{\sqrt{2}\sqrt{2}}=\frac{1}{2}\Rightarrow \theta =60°$$

Worked example (perpendicular). For $2\hat{i}+\lambda \hat{j}+\hat{k}$ to be perpendicular to $\hat{i}-2\hat{j}+3\hat{k}$, we need $2-2\lambda +3=0$, so $\lambda =2.5$.

Worked example (magnitudes). If $|\vec{a}|=3$, $|\vec{b}|=4$ and $\vec{a}\cdot \vec{b}=6$, then $|\vec{a}+\vec{b}{|}^2=9+12+16=37$, so $|\vec{a}+\vec{b}|=\sqrt{37}\approx 6.08$.

An everyday example. Checking that a shelf bracket meets the wall at a right angle is the physical version of testing that a dot product is zero.

The substance. A zero dot product does not mean one vector is zero — non-zero perpendicular vectors give zero as well.

**The scalar projection of $\vec{a}$ on $\vec{b}$ is $\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$, the signed length of its shadow along $\vec{b}$, and the vector projection is that length times the unit vector $\hat{b}$, namely $\frac{\vec{a}\cdot \vec{b}}{|\vec{b}{|}^2}\vec{b}$.

Worked example.** Find the projections of $\vec{a}=2\hat{i}+3\hat{j}+2\hat{k}$ on $\vec{b}=\hat{i}+2\hat{j}+2\hat{k}$.

- $\vec{a}\cdot \vec{b}=2+6+4=12$ and $|\vec{b}|=3$
- Scalar projection: $\frac{12}{3}=4$
- Vector projection: $\frac{12}{9}(\hat{i}+2\hat{j}+2\hat{k})=\frac{4}{3}\hat{i}+\frac{8}{3}\hat{j}+\frac{8}{3}\hat{k}$

Check. What is left of $\vec{a}$ after removing the vector projection is $\frac{2}{3}\hat{i}+\frac{1}{3}\hat{j}-\frac{2}{3}\hat{k}$, and its dot product with $\vec{b}$ is $\frac{2}{3}+\frac{2}{3}-\frac{4}{3}=0$ — it is perpendicular to $\vec{b}$, as it should be.

An everyday example. The shadow of a leaning bamboo pole on flat ground when the sun is overhead is its projection on the ground, and the scalar projection gives the shadow's length.

The substance. **The projection of $\vec{a}$ on $\vec{b}$ differs from that of $\vec{b}$ on $\vec{a}$** — here the second is $\frac{12}{\sqrt{17}}\approx 2.91$, not 4.