Last updated: 2026-06-04

What is the dot product?

The dot product (also called the scalar product or inner product) is an operation that takes two vectors of equal dimension and returns a single number. For vectors a = (a₁, a₂, …, aₙ) and b = (b₁, b₂, …, bₙ), it is defined as:

a · b = a₁b₁ + a₂b₂ + … + aₙbₙ = Σᵢ aᵢbᵢ

This algebraic definition is straightforward to compute from components, but the dot product also has a geometric interpretation that makes it indispensable in physics and computer graphics.

Geometric interpretation

The dot product connects algebra to geometry through the angle θ between the two vectors:

a · b = |a| |b| cos θ

where |a| and |b| are the magnitudes (lengths) of the vectors. Rearranging gives the angle formula used in this calculator:

θ = arccos(a · b / (|a| |b|))

This tells us immediately what the sign of the dot product means:

• Positive (θ < 90°): the vectors lean toward each other
• Zero (θ = 90°): the vectors are perpendicular (orthogonal)
• Negative (θ > 90°): the vectors lean away from each other

Worked example

Consider a = (2, 3, −1) and b = (4, −1, 2).

Step 1 — compute the dot product: a · b = (2)(4) + (3)(−1) + (−1)(2) = 8 − 3 − 2 = 3

Step 2 — compute the magnitudes: |a| = √(4 + 9 + 1) = √14 ≈ 3.7417 |b| = √(16 + 1 + 4) = √21 ≈ 4.5826

Step 3 — find the angle: cos θ = 3 / (√14 × √21) = 3 / √294 ≈ 0.1750 θ = arccos(0.1750) ≈ 79.92°

The two vectors are nearly perpendicular but not quite — they share a small positive component along the same direction.

Orthogonality

Two vectors are orthogonal when their dot product is exactly zero, meaning they are perpendicular regardless of dimension. Standard basis vectors (1, 0, 0) and (0, 1, 0) are orthogonal; so are (3, 4) and (−4, 3) in two dimensions. Orthogonality is a cornerstone of signal processing (Fourier components), quantum mechanics (state superposition), and machine learning (independent features).

Vector projection

One of the most practical uses of the dot product is computing projections. The scalar projection of a onto b — how much of a lies along b — is:

proj_scalar = a · b / |b|

The corresponding vector projection (a vector in the direction of b):

proj_vector = (a · b / |b|²) b

For example, if a = (5, 3) and b = (1, 0), the scalar projection is 5 and the vector projection is (5, 0) — the horizontal component of a.

Applications

The dot product appears across science and engineering:

• Work in physics: W = F · d (force dotted with displacement)
• Lighting in 3D graphics: the brightness of a surface is proportional to the dot product of the surface normal with the light direction
• Cosine similarity in text analysis and recommendation systems: measuring how aligned two documents or preference vectors are
• Least-squares regression: the normal equations minimise ‖Ax − b‖ by setting dot products of residuals with basis vectors to zero
• Gram–Schmidt orthogonalisation: successive projections are used to construct orthogonal bases from arbitrary vectors

Relationship to the cross product

For three-dimensional vectors, the dot product a · b = |a| |b| cos θ captures the component of alignment, while the cross product a × b has magnitude |a| |b| sin θ and captures the component of perpendicularity (area of the parallelogram spanned by the two vectors). Together they characterise the full relationship between any two 3D vectors.