Dot Product

The inner product of two vectors — a scalar measuring alignment, enabling projections, angles, and the geometry of vector spaces.

The dot product measures how much two vectors point in the same direction

Formula

a dot b = ||a||||b||cos(theta)

a dot b = 4.63

Positive means aligned. Zero means perpendicular. Negative means pointing against each other.

theta = 50°

Definition

The dot product (inner product) of two vectors $\mathbf{a} = (a_1, \ldots, a_n)$ and $\mathbf{b} = (b_1, \ldots, b_n)$ is:

$\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^n a_i b_i = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n$

The result is a scalar (single number), not a vector.

Geometric meaning: $\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta$ where $\theta$ is the angle between the vectors.

Key properties

Commutative: $\mathbf{a}\cdot\mathbf{b} = \mathbf{b}\cdot\mathbf{a}$
Distributive over addition: $\mathbf{a}\cdot(\mathbf{b}+\mathbf{c}) = \mathbf{a}\cdot\mathbf{b} + \mathbf{a}\cdot\mathbf{c}$
Compatible with scalars: $(c\mathbf{a})\cdot\mathbf{b} = c(\mathbf{a}\cdot\mathbf{b})$
$\mathbf{a}\cdot\mathbf{a} = \|\mathbf{a}\|^2 \geq 0$ , with equality only when $\mathbf{a} = \mathbf{0}$

Common mistakes

Confusing the dot product with the cross product: the dot product yields a scalar and works in any dimension; the cross product yields a vector and is specific to 3D
Dimension mismatch: the dot product is only defined for vectors of the same length — $(1,2,3)\cdot(1,2)$ is meaningless

Computing dot products

$\mathbf{a} = (3, -1, 2)$ , $\mathbf{b} = (1, 4, -2)$ .

$\mathbf{a} \cdot \mathbf{b} = 3(1) + (-1)(4) + 2(-2) = 3 - 4 - 4 = -5$ .

Since the result is negative, the vectors point in roughly opposite directions.

Try it

Find the dot product of $(2, 3)$ and $(-3, 2)$ . Are these vectors perpendicular?

Solution

$(2)(-3) + (3)(2) = -6 + 6 = 0$ . Yes, they are perpendicular — the dot product is zero, confirming $\theta = 90°$ .

This pair is a 90° rotation of each other — a general fact: $(a, b)$ and $(-b, a)$ are always perpendicular.

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