Are dot product, scalar product, and inner product the same?

In this context, yes. Each means adding the products of corresponding components of equal-dimensional vectors, producing a scalar result.

Why is the cross product limited to three dimensions?

The standard cross product produces a three-dimensional vector perpendicular to two three-dimensional inputs. This calculator uses that classical definition.

How do I add two vectors?

Add components in matching positions of equal-dimensional vectors. The result is a new vector of the same dimension.

How is vector magnitude calculated?

Square the components, add the squares, and take the square root. This is also called the Euclidean norm.

How do I calculate a unit vector?

Divide every component of a nonzero vector by its magnitude. The result points in the same direction and has magnitude 1.

How do I find the angle between two vectors?

Divide their dot product by the product of their magnitudes, then take the inverse cosine.

What does vector projection represent?

The projection of A onto B is the vector component of A that lies in the direction of B.

Why can a zero vector not be normalized?

Normalization divides a vector by its magnitude. The zero vector has magnitude zero, so that division is undefined.

Tools/🧮 Calculators/Vector Calculator

Vector Calculator

Add and subtract vectors or calculate dot product, cross product, magnitude, unit vector, angle, and projection.

Version 1.0 · Last Updated: August 10, 2026 · Maintained by ToolLarder

Online vector calculations and vector operations

Perform common linear algebra operations on vectors from 2 to 6 dimensions. Calculate dot product, magnitude, unit vector, angle, and projection, plus the three-dimensional cross product.

Vector addition and subtraction

Add or subtract corresponding components of vectors with equal dimensions.

A ± B = (a₁ ± b₁, …, aₙ ± bₙ)

Dot product (scalar product / inner product)

Multiply corresponding components and add the products to produce one scalar value.

A · B = Σ aᵢbᵢ

Cross product (vector product)

Calculate a new vector perpendicular to two three-dimensional vectors.

A × B = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁)

Vector magnitude (norm)

The square root of the sum of squared components gives the Euclidean norm.

‖A‖ = √Σaᵢ²

Unit vector calculator

Divide a nonzero vector by its magnitude to create a length-one vector in the same direction.

Â = A / ‖A‖

Angle between two vectors

Use the dot product and vector magnitudes to find the angle in degrees and radians.

θ = arccos((A·B)/(‖A‖‖B‖))

Vector projection calculator

Find the component of vector A that lies in the direction of vector B.

projᴮ(A) = ((A·B)/(B·B))B

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