The Dot Product and the Angle Between Vectors

Multiplying two vectors into a number

Adding vectors gives a vector. Multiplying a vector by a number gives a vector. The dot product does something else: it turns two vectors into a single number.

$$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2$$

For $\mathbf{u} = \langle 3, 4\rangle$ and $\mathbf{v} = \langle 2, -1\rangle$,

$$\mathbf{u} \cdot \mathbf{v} = 3(2) + 4(-1) = 2$$

That number carries geometric information: it measures how much the two vectors point the same way.

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

where $\theta$ is the angle between them, taken between $0°$ and $180°$.

Why the components know the angle

The three arrows above form a triangle with sides $|\mathbf{u}|$, $|\mathbf{v}|$ and $|\mathbf{u} - \mathbf{v}|$, and the angle $\theta$ sits between the first two. The law of cosines says

$$|\mathbf{u} - \mathbf{v}|^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 - 2|\mathbf{u}||\mathbf{v}|\cos\theta$$

Now expand the left side in components:

$$(u_1 - v_1)^2 + (u_2 - v_2)^2 = \left(u_1^2 + u_2^2\right) + \left(v_1^2 + v_2^2\right) - 2\left(u_1v_1 + u_2v_2\right)$$

The first two brackets are $|\mathbf{u}|^2$ and $|\mathbf{v}|^2$, so comparing the two lines leaves $u_1v_1 + u_2v_2 = |\mathbf{u}||\mathbf{v}|\cos\theta$. The component formula and the angle formula are the law of cosines written twice, which is why a sum of products can report an angle.

Two consequences follow at once. Since $|\mathbf{u}|$ and $|\mathbf{v}|$ are positive, the sign of the dot product is the sign of $\cos\theta$:

Dot product	Angle
positive	acute, under $90°$
zero	right angle
negative	obtuse, over $90°$

And taking $\mathbf{v} = \mathbf{u}$ gives $\mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2$.

Projections

The dot product also answers: how far does $\mathbf{u}$ reach in the direction of $\mathbf{v}$? Dropping a perpendicular from the tip of $\mathbf{u}$ onto the line through $\mathbf{v}$ leaves a shadow of length $|\mathbf{u}|\cos\theta$, which is the dot product divided by $|\mathbf{v}|$:

$$\text{comp}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|} \qquad \text{proj}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2}\,\mathbf{v}$$

The first is a number, the length of the shadow. The second is the shadow itself, a vector along $\mathbf{v}$.

Worked examples

Common mistakes

Practice problems