Dot product

The dot product of two vectors is a scalar that measures how much they point in the same direction.

Two definitions

Algebraic (component-wise):

$\mathbf{A}\cdot \mathbf{B}=a_x{b}_x+a_y{b}_y+a_z{b}_z.$

Geometric: let $\theta$ be the angle between $\mathbf{A}$ and $\mathbf{B}$, $0\le \theta \le \pi$. Then

$\mathbf{A}\cdot \mathbf{B}=|\mathbf{A}||\mathbf{B}|\cos \theta .$

The two definitions agree. Proof: apply the law of cosines to the triangle with sides $\mathbf{A},\mathbf{B},\mathbf{A}-\mathbf{B}$; the algebraic expression for $|\mathbf{A}-\mathbf{B}{|}^2=(\mathbf{A}-\mathbf{B})\cdot (\mathbf{A}-\mathbf{B})$ matches the geometric one $|\mathbf{A}{|}^2+|\mathbf{B}{|}^2-2|\mathbf{A}||\mathbf{B}|\cos \theta$ after expansion.

What it tells you

The geometric definition is the source of the dot product’s usefulness. The sign of $\mathbf{A}\cdot \mathbf{B}$ alone classifies the angle:

• Positive: acute ($0\le \theta <\pi /2$). Vectors point “mostly the same way.”
• Zero: perpendicular ($\theta =\pi /2$). The workhorse for orthogonality checks.
• Negative: obtuse ($\pi /2<\theta \le \pi$). Vectors point “mostly opposite ways.”

Two nonzero vectors are perpendicular iff their dot product is zero. This is one of the most-used facts in vector calculus.

Properties

• $\mathbf{A}\cdot \mathbf{A}=|\mathbf{A}{|}^2$ (magnitude squared).
• $\mathbf{A}\cdot \mathbf{B}=\mathbf{B}\cdot \mathbf{A}$ (commutative).
• $\mathbf{A}\cdot (\mathbf{B}+\mathbf{C})=\mathbf{A}\cdot \mathbf{B}+\mathbf{A}\cdot \mathbf{C}$ (distributive).
• $(c\mathbf{A})\cdot \mathbf{B}=c(\mathbf{A}\cdot \mathbf{B})$.
• Standard unit vectors: $\hat{\mathbf{i}}\cdot \hat{\mathbf{i}}=1$, $\hat{\mathbf{i}}\cdot \hat{\mathbf{j}}=0$, etc.

The component formula falls right out of these properties together with the orthonormality of $\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}$.

Angle and projection

Solve the geometric formula for $\cos \theta$:

$\cos \theta =\frac{\mathbf{A}\cdot \mathbf{B}}{|\mathbf{A}||\mathbf{B}|}.$

The scalar projection of $\mathbf{v}$ onto $\hat{\mathbf{u}}$ is $\mathbf{v}\cdot \hat{\mathbf{u}}$ — the (signed) component of $\mathbf{v}$ in the direction $\hat{\mathbf{u}}$.

The vector projection is the actual shadow vector:

${\mathrm{proj}}_{\mathbf{u}}\mathbf{v}=\left(\frac{\mathbf{u}\cdot \mathbf{v}}{|\mathbf{u}{|}^2}\right)\mathbf{u}.$

In physics and engineering

Work done by a force $\mathbf{F}$ moving an object through displacement $\mathbf{d}$: $W=\mathbf{F}\cdot \mathbf{d}$. The dot product picks out the force component along motion. Perpendicular force components contribute nothing.

Power delivered by a force at velocity $\mathbf{v}$: $P=\mathbf{F}\cdot \mathbf{v}$.

Flux through a flat surface with constant field $\mathbf{F}$ and unit normal $\hat{\mathbf{N}}$: $\mathbf{F}\cdot \hat{\mathbf{N}}\cdot \text{Area}$. The dot product extracts the component of the field that’s actually crossing the surface.

In vector calculus

The dot product is the integrand in the vector line integral ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ — “work done by $\mathbf{F}$ along $C$.” It’s also the integrand in the Flux integral ${\iint }_S\mathbf{F}\cdot d\mathbf{S}$. The recurring theme: dot products convert vector quantities into scalar quantities that respect orientation.

The divergence $\nabla \cdot \mathbf{F}$ is, formally, a “dot product” of the differential operator $\nabla$ with the vector field $\mathbf{F}$. See Divergence.