Adding vectors gives a vector. A different product of two vectors gives a single number that measures how much they point the same way. It is the key to angles between vectors.
Key idea
$$\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd \qquad\text{and}\qquad \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta,$$ where $\theta$ is the angle between them. So $\cos\theta = \dfrac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$, and $\mathbf{u}$ and $\mathbf{v}$ are perpendicular exactly when $\mathbf{u} \cdot \mathbf{v} = 0$.
Why it works
Put $\mathbf{u}$ and $\mathbf{v}$ tail to tail; the third side of the triangle is $\mathbf{u} - \mathbf{v}$. The law of cosines says $$|\mathbf{u} - \mathbf{v}|^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 - 2|\mathbf{u}||\mathbf{v}|\cos\theta.$$ In components the left side is $$(a - c)^2 + (b - d)^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 - 2(ac + bd).$$ Comparing the two gives $ac + bd = |\mathbf{u}||\mathbf{v}|\cos\theta$.
Worked examples
Worked example: Computing
Find $\langle 3, -2 \rangle \cdot \langle 4, 5 \rangle$.
- $3 \cdot 4 + (-2) \cdot 5 = 12 - 10$.
Answer: $2$ (a number, not a vector)
Worked example: An angle
Find the angle between $\mathbf{u} = \langle 1, 1 \rangle$ and $\mathbf{v} = \langle -1, 0 \rangle$.
- $\mathbf{u} \cdot \mathbf{v} = -1$; $|\mathbf{u}| = \sqrt{2}$, $|\mathbf{v}| = 1$.
- $\cos\theta = \dfrac{-1}{\sqrt{2}} = -\dfrac{\sqrt{2}}{2}$.
Answer: $\theta = 135^\circ$
Worked example: Perpendicular
Find $k$ so that $\langle k, 3 \rangle$ is perpendicular to $\langle 2, -4 \rangle$.
- Set the dot product to $0$: $2k - 12 = 0$.
Answer: $k = 6$
Common mistake
Treating the dot product as a vector, for example writing $\langle 3, -2 \rangle \cdot \langle 4, 5 \rangle = \langle 12, -10 \rangle$. Multiply matching components and then add; the result is one number. Its sign alone tells you whether the angle is acute ($+$), right ($0$) or obtuse ($-$).