What is the dot product of two 3D vectors?

For vectors A = (a1, a2, a3) and B = (b1, b2, b3), the dot product is A·B = a1·b1 + a2·b2 + a3·b3. The result is a scalar — a single number, not a vector.

How is the angle between vectors calculated?

Using the relation A·B = |A| × |B| × cos(θ), you can isolate θ: θ = arccos((A·B) / (|A| × |B|)). This tool computes that angle in both degrees and radians.

What does a dot product of zero mean?

A dot product of exactly zero means the two vectors are orthogonal — perpendicular to each other at a 90-degree angle.

Can I use negative or decimal components?

Yes. All component fields accept any real number — positive, negative, or decimal. Results are rounded to 6 significant figures for readability.

What is the difference between the dot product and the cross product?

The dot product produces a scalar and measures alignment (cosine similarity). The cross product produces a new vector perpendicular to both inputs and measures the sine of the angle. Use the cross product calculator on this site for that operation.

What if one of the vectors is the zero vector?

If either vector is (0, 0, 0), its magnitude is zero and the angle is undefined (division by zero). The tool shows "undefined" for the angle in that case.

3D Dot Product Calculator

Enter two 3D vectors and instantly get their dot product, magnitudes, and the angle between them.

Enter 3D Vectors

Vector A

Vector B

Quick Examples

Enter two vectors and click Calculate

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Summary

The dot product of two 3D vectors A and B is calculated as a1·b1 + a2·b2 + a3·b3, producing a single scalar. This tool also derives both vector magnitudes and the angle between the vectors using the formula cos(θ) = (A·B) / (|A| × |B|), making it useful for physics, computer graphics, and linear algebra.

How it works

Enter the x, y, and z components for Vector A.
Enter the x, y, and z components for Vector B.
The dot product, magnitudes, and angle update instantly as you type.
Read the step-by-step breakdown to verify each multiplication.
Use the Reset button to clear all fields and start over.

Use cases

Calculate work done by a 3D force along a displacement vector (W = F · d).
Find the angle between two surfaces or edges in 3D modeling.
Test orthogonality: a dot product of 0 confirms vectors are perpendicular.
Compute lighting intensity in shading equations (normal · light direction).
Verify vector projections and decomposition in physics problems.
Check correlation between two feature vectors in machine learning.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu