What is the dot product?

The dot product is a way to multiply two vectors to get a scalar (single number). It measures how much two vectors point in the same direction.

When is the dot product zero?

The dot product is zero when the vectors are perpendicular (orthogonal). This is a common test for perpendicularity.

What is the difference between dot product and cross product?

The dot product gives a scalar and measures alignment. The cross product gives a vector perpendicular to both inputs and measures the area of the parallelogram they span.

Dot Product Calculator

Calculate the dot product of two 3D vectors. Also shows the angle between them and scalar projection.

Save

Vector A: x

Vector A: y

Vector A: z

Vector B: x

Vector B: y

Vector B: z

Dot

32.0000

Mag A3.7417

Mag B8.7750

Angle12.9332 deg

View formula & how it works →

Dot Product

Formula

A · B = a₁b₁ + a₂b₂ + a₃b₃

Geometric Interpretation

A · B = A×B× cos(theta)

So: theta = arccos(A · B / (A×B))

Properties

If A · B = 0, the vectors are perpendicular

If A · B > 0, the angle is acute (less than 90°)

If A · B < 0, the angle is obtuse (greater than 90°)

Example Calculation

Dot product of (1,2,3) and (4,5,6).

01A · B = 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32
02|A| = √(1+4+9) = √14 ≈ 3.742
03|B| = √(16+25+36) = √77 ≈ 8.775
04cos(theta) = 32/(3.742 × 8.775) ≈ 0.9746
05theta ≈ 12.93°

Frequently Asked Questions

Learn More

How to Calculate Percentages - Complete Guide

Learn how to calculate percentages step by step. Covers finding a percentage of a number, percentage change, reverse percentages, and real-world applications.

Dot Product Calculator

Dot Product

Formula

Geometric Interpretation

Properties

Example Calculation

Frequently Asked Questions

How to Calculate Percentages - Complete Guide

Related Calculators

Cross Product Calculator

Vector Magnitude Calculator

Matrix Determinant Calculator