What is the difference between the dot and cross products?

The dot product is a number, |a||b|cosθ, used for angles and projections. In 3D the cross product is a vector perpendicular to both, whose magnitude is the area of the parallelogram they span.

How do I find the angle between two vectors?

cosθ = a·b / (|a||b|), then take the inverse cosine. Degrees and radians are both shown, and a zero vector is reported as undefined.

How do I check whether vectors are perpendicular or parallel?

A dot product of 0 means perpendicular; a zero cross product (in 2D, x₁y₂−x₂y₁ = 0) means parallel. The tool states the conclusion for you.

What is the scalar projection?

The length of a along b: a·b / |b|. Multiplying by the unit vector of b gives the vector projection, and both are computed.

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➡️Vector Calculator - Dot Product, Cross Product, Magnitude and Angle

Works with 2D and 3D vectors: a+b, a−b, k·a, the dot and cross products, both magnitudes, the angle between them, the unit vector of a, the scalar and vector projection of a onto b, and whether the vectors are perpendicular or parallel. The cross product magnitude also gives the parallelogram and triangle areas.

Free · No sign-up · Runs in your browserLast updated: 2026-10-26

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How to use Vector Calculator

1Choose 2D or 3D
2Enter the components of a and b, and a scalar k if you need k·a
3Read the results and copy any component

FAQ

What is the difference between the dot and cross products?: The dot product is a number, |a||b|cosθ, used for angles and projections. In 3D the cross product is a vector perpendicular to both, whose magnitude is the area of the parallelogram they span.
How do I find the angle between two vectors?: cosθ = a·b / (|a||b|), then take the inverse cosine. Degrees and radians are both shown, and a zero vector is reported as undefined.
How do I check whether vectors are perpendicular or parallel?: A dot product of 0 means perpendicular; a zero cross product (in 2D, x₁y₂−x₂y₁ = 0) means parallel. The tool states the conclusion for you.
What is the scalar projection?: The length of a along b: a·b / |b|. Multiplying by the unit vector of b gives the vector projection, and both are computed.