Dot product calculator calculates the dot product of two vectors ** a** and

**in Euclidean space**

*b***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

**dot**

*a*

*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.

*(a _{i }a_{j} a_{k}) ∙ (b_{i} b_{j} b_{k}) = (a_{i} ∙ b_{i} + a_{j} ∙ b_{j} + a_{k} ∙ b_{k})*

Where

** i, j,** and

**refers to**

*k***and**

*x, y,***coordinates on Cartesian plane.**

*z*## 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.