Dot product Calculator

Formula first

Overview

The dot product, also known as the scalar product, is an algebraic operation that takes two vectors and returns a single scalar value. Geometrically, it represents the product of the magnitudes of the two vectors and the cosine of the angle between them, quantifying how much one vector aligns with the other.

Symbols

Variables

|a| = Magnitude of a, |b| = Magnitude of b, \theta = Angle θ, \mathbf{a}\cdot\mathbf{b} = Dot Product

Apply it well

When To Use

When to use: Use this formula when you need to calculate the angle between two vectors or find the projection of one vector onto another. It is the primary method for determining if two vectors are orthogonal, as their dot product will be exactly zero in such cases.

Why it matters: In physics, the dot product is used to calculate work done by a force over a displacement. In computer science, it is fundamental for 3D graphics shading, machine learning similarity scores, and signal processing.

Avoid these traps

Common Mistakes

• Using sine instead of cosine.
• Confusing with cross product.

One free problem

Practice Problem

A force vector has a magnitude of 10 and a displacement vector has a magnitude of 5. If the angle between them is 60°, find the resulting dot product.

Solve for: $dot$

Hint: The cosine of 60° is 0.5.

The full worked solution stays in the interactive walkthrough.

Keep going

Related Formulas

References

Sources

1. Halliday, Resnick, and Walker, Fundamentals of Physics
2. Wikipedia: Dot product
3. Bird, Stewart, and Lightfoot, Transport Phenomena
4. Stewart, James. Calculus: Early Transcendentals. 8th ed. Cengage Learning, 2016.
5. Anton, Howard, and Chris Rorres. Elementary Linear Algebra: Applications Version. 11th ed. Wiley, 2013.
6. Standard curriculum — A-Level Pure Mathematics (Vectors)