Math & Business Productivity

Vector Dot & Cross Product Calculator

Calculate the dot product, cross product, magnitudes, and angle between two 3D vectors. Free vector mathematics and scalar calculator.

Formula Verified & Maintained by:Dr. Marcus Vance, Ph.D.

Ph.D. Applied Mathematics (MIT) • Lead Mathematician & Psychometric Systems Editor

Peer-Reviewed AlgorithmAudited for 2026 Rules

Interactive Calculator

1. Provide Details

Vector A - X Component (Ax)

Vector A - Y Component (Ay)

Vector A - Z Component (Az)

Vector B - X Component (Bx)

Vector B - Y Component (By)

Vector B - Z Component (Bz)

2. Output Results

Dot Product (Scalar Product)-41

Cross Product (Vector Product)[-13, 17, 31]

Magnitude of Vector A |A|7.071

Magnitude of Vector B |B|7.874

Angle Between Vectors137.4 degrees

How to Calculate: Formula & Steps

Computes the scalar dot product by summing the products of corresponding components, and computes the 3D cross product vector perpendicular to both input vectors.

Formula Used:A · B = AxBx + AyBy + AzBz; A × B = [AyBz - AzBy, AzBx - AxBz, AxBy - AyBx]

Step-by-Step Calculation Example

For A = [3, -5, 4] and B = [2, 7, -3]: Dot Product is -41; Cross Product is [-13, 17, 31].

Common Mistakes to Avoid

Confusing the dot product (which outputs a single number) with the cross product (which outputs a new 3D vector).
Attempting to calculate a cross product for 2D vectors directly (the cross product is only defined in three dimensions, though 2D vectors can be represented in 3D by setting the Z component to 0).

Practical Use Cases

Physics force and torque calculations
3D computer graphics and game development
Linear algebra homework checking
Engineering structural analysis

Expert Tips

To calculate 2D vectors, simply leave the Z component (Az and Bz) empty or set to 0.
The order of vectors matters for the cross product: A × B is the negative (opposite direction) of B × A.

3D Vector Algebra: Dot Product, Cross Product & Orthogonality

In physics, linear algebra, and 3D computer graphics, vector multiplication is defined under two distinct operations: the Dot Product (Scalar Product) and the Cross Product (Vector Product).

Dot Product (Scalar Output)

A · B = AxBx + AyBy + AzBz = |A||B| cos(θ)

Produces a single scalar real number. If A · B = 0, the two vectors are perpendicular (orthogonal at 90°).

Cross Product (3D Vector Output)

A × B = [AyBz - AzBy, AzBx - AxBz, AxBy - AyBx]

Produces a brand-new 3D vector perpendicular (normal) to both input vectors with magnitude equal to |A||B| sin(θ).

Angle Between Two 3D Vectors Formula

θ = arccos[ (A · B) ÷ (|A| × |B|) ]

Where |A| = √(Ax² + Ay² + Az²) and |B| = √(Bx² + By² + Bz²).

Disclaimer

This calculation tool is provided for educational and informational estimation purposes only. Results are based on mathematical formulas and user-supplied parameters. They do not constitute formal underwriting, financial, tax, engineering, or legal determinations.

Frequently Asked Questions About Vector Dot & Cross Product Calculator

What is a dot product (scalar product)?

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single scalar number. It measures how much the vectors point in the same direction.

What is a cross product (vector product)?

The cross product is a binary operation on two vectors in a three-dimensional space. It results in a third vector that is perpendicular (orthogonal) to both of the original vectors, with a magnitude equal to the area of the parallelogram that the vectors span.

How do you calculate the dot product of two vectors?

Multiply the corresponding components of the two vectors together and sum the results: A · B = (Ax × Bx) + (Ay × By) + (Az × Bz).

How do you calculate the cross product of two vectors?

Use the determinant formula for 3D vectors: AxB = [ (Ay × Bz - Az × By), (Az × Bx - Ax × Bz), (Ax × By - Ay × Bx) ]. The result is always a 3D vector, not a scalar.

What is the angle between perpendicular and parallel vectors?

If the dot product of two vectors is exactly 0, the vectors are perpendicular (90 degrees). If the cross product of two vectors is [0, 0, 0], the vectors are parallel (0 or 180 degrees).