Dot product calculator calculates the dot product of two vectors a and b in Euclidean space. Enter i, j, and k for both vectors to get scalar number.

a . b

Vector dot product calculator shows step by step scalar multiplication.

## What is dot product?

Dot product is an algebraic operation that takes two equal-length sequences of numbers usually coordinate vectors, and returns a single number.

Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.

a . b usually read as a dot b.

Image credit: “Dot Product” by Math is Fun.

## Dot product formula

### Vector Magnitude

Use this equation to calculate dot product of two vectors if magnitude (length) is given.

a ∙ b = |a| × |b| × cos(θ)

Where

|a| is length of vector a

|b| is length of vector b

θ is the angle between a and b

### Vector Directions

We can also find dot product by using the direction of both vectors.

(ai aj ak) ∙ (bi bj bk) = (ai ∙ bi + aj ∙ bj + ak ∙ bk)

Where

i, j, and k refers to x, y, and z coordinates on Cartesian plane.

## How to find dot product of two vectors?

Dot product of two vectors can calculated by using the dot product formula.

Method 1 – Vector Direction

Vector a = (2i, 6j, 4k)

Vector b = (5i, 3j, 7k)

Place the values in the formula.

a ∙ b = (2, 6, 4) ∙ (5, 3, 7)

(ai aj ak) ∙ (bi bj bk) = (ai ∙ bi + aj ∙ bj + ak ∙ bk)
(2   6   4) ∙ (5   3   7) = (2 ∙ 5 + 6 ∙ 3 + 4 ∙ 7)
(2   6   4) ∙ (5   3   7) = (10 + 18 + 28)

a ∙ b = 56

Method 2 – Vector Magnitude

|a| = 15, |b| = 10, θ = 30°

Place the values in the formula.

a · b = |a| × |b| × cos(θ)

a · b = 15 × 10 × cos(30°)

a · b = 23.14

References:

Dot Product Formula from tutorial.math.lamar.edu.