What is the dot product of two vectors?

The dot product (or scalar product) of two vectors a and b in n-dimensional space is the sum of the products of their corresponding components: a·b = a1b1 + a2b2 + ... + anbn. Geometrically, it can also be written as a·b = |a||b|cos(θ), where θ is the angle between the two vectors.

How do I compute the angle between two vectors from the dot product?

Once you know the dot product a·b and the magnitudes |a| and |b|, you can compute the cosine of the angle θ as cos(θ) = (a·b) / (|a||b|). Then θ = arccos( cos(θ) ). If one of the vectors is the zero vector, the angle is undefined because its direction is not defined.

How can I tell if two vectors are orthogonal using the dot product?

Two vectors are orthogonal (perpendicular) if and only if their dot product is zero. In numerical computations, small rounding errors are common, so we usually check if the dot product is very close to zero within a tolerance rather than exactly 0.

What is the projection of one vector onto another?

The projection of a vector a onto a non-zero vector b is the vector proj_b(a) = [(a·b)/|b|^2] b. It represents the component of a that points in the direction of b. The length of this projection is |a|cos(θ), where θ is the angle between the vectors.

Can this dot product calculator handle more than 3 dimensions?

Yes. The calculator accepts coordinate lists like 1, 2, 3, 4 or 0.5 -1.3 2.7, so you can work in 2D, 3D, or any n-dimensional space as long as the two vectors have the same number of components.

Dot Product Calculator – Vectors, Angle Between Vectors & Projection

This dot product calculator works with 2D, 3D, and general n-dimensional vectors. Enter the components of two vectors and get the scalar product a·b, the length of each vector, the angle between them, and the projection of A onto B, with step-by-step expansion.

It is designed for students and professionals in linear algebra, physics, engineering, and data science who need a precise and readable breakdown of vector computations.

Dot product – algebraic and geometric definitions

For two vectors \( \mathbf{a}, \mathbf{b} \in \mathbb{R}^n \) with components \( \mathbf{a} = (a_1, a_2, \dots, a_n) \) and \( \mathbf{b} = (b_1, b_2, \dots, b_n) \), the dot product (or scalar product) is defined as

\[ \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = a_1 b_1 + a_2 b_2 + \dots + a_n b_n. \]

Geometrically, the dot product relates to the angle \( \theta \) between the two vectors:

\[ \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \, \|\mathbf{b}\| \cos\theta, \]

where \( \|\mathbf{a}\| \) and \( \|\mathbf{b}\| \) denote the Euclidean norms (lengths). This gives

\[ \cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \, \|\mathbf{b}\|}, \qquad \theta = \arccos\!\left(\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \, \|\mathbf{b}\|}\right). \]

Orthogonality and angle between vectors

Orthogonal vectors: \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if and only if \( \mathbf{a} \cdot \mathbf{b} = 0 \). Geometrically, the angle between them is \( 90^\circ \).
Parallel vectors: If \( \theta = 0^\circ \) or \( 180^\circ \), the vectors are parallel or anti-parallel. In practice we test whether the unit vectors have the same or opposite direction within a tolerance.
Zero vector: If one of the vectors is zero, its direction is undefined, so the angle is also undefined even though the dot product is 0.

Projection of one vector onto another

The projection of \( \mathbf{a} \) onto a non-zero vector \( \mathbf{b} \) is the component of \( \mathbf{a} \) that lies in the direction of \( \mathbf{b} \). It is given by

\[ \operatorname{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2}\,\mathbf{b}. \]

The scalar factor \( (\mathbf{a} \cdot \mathbf{b}) / \|\mathbf{b}\|^2 \) tells you how many “copies” of \( \mathbf{b} \) you need to reach the shadow of \( \mathbf{a} \) along \( \mathbf{b} \). The length of this projection is \( \|\operatorname{proj}_{\mathbf{b}}(\mathbf{a})\| = \|\mathbf{a}\|\cos\theta \).

Typical applications of the dot product

Work in physics: Work done by a constant force \(\mathbf{F}\) along a displacement \(\mathbf{s}\) is \( W = \mathbf{F} \cdot \mathbf{s} = \|\mathbf{F}\|\|\mathbf{s}\|\cos\theta \).
Projections and decompositions: Splitting a vector into components along and orthogonal to a direction.
Similarity in data science: The cosine of the angle between feature vectors is a measure of similarity used in information retrieval and recommender systems.
Linear algebra: Orthogonality, orthogonal projections, and orthonormal bases are all built on the dot product.

Dot Product Calculator

1. Enter vectors

2. Results

Vector summary

Projection of A onto B

Step-by-step expansion (component form)

Dot product – algebraic and geometric definitions

Orthogonality and angle between vectors

Projection of one vector onto another

Typical applications of the dot product

Dot product – FAQ