# Dot Product Calculator

Enter values to find the online calculator dot product of two vectors with the dot product calculator.

Define & calculating each vector

Vector a:
i:
j:
k:

Vector b:
i:
j:
k:

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

A Dot Product Calculator is a tool that computes the dot product (also known as scalar product or inner product) of two vectors in Euclidean space. The dot product is a scalar value that represents the extent to which two vectors are aligned. It has numerous applications in geometry, physics, and engineering.

To use the dot product calculator, you need to enter the components (component) i, j, and k for both vectors, which are generally represented as a = (a1, a2, a3) and b = (b1, b2, b3). These components correspond to the x, y, and z dimensions in a three-dimensional Euclidean space ( second vector).

Scalar Product of Two Vectors

The dot product calculator computes this scalar value for you, given the components i, j, and k for both vectors. This scalar number can then be used for various purposes, such as determining the angle between two vectors, testing if vectors are orthogonal (dot product equals zero), or finding the projection of one vector onto another.

In summary, a Dot Product Calculator simplifies the process of finding the dot product of two vectors in Euclidean space by requiring only the i, j, and k components of both vectors to calculate the 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.

## Dot product formula (two vectors)

Use this equation to calculate dot product of two vectors if magnitude (length) is given. Vector Multiplication calculation for vector components

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.

(aaj 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?

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

Dot Product Example 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)

Solution – a ∙ b = 56

Dot Product Example 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°)

Solution – a · b = 23.14

References:

Two non-zero vectors are perpendicular if and only if their scalar product equals to zero

Dot product of two vectors a and b is a scalar quantity equal to the sum of pairwise products of coordinate vectors a and b

Dot Product Cross Product Magnitude Angle Unit Projection Scalar Projection Orthogonal Projection