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# Dot Product Calculator

Input the values of vectors a & b and click the calculate button to find a.b using dot product calculator

## Dot product calculator

Dot product calculator is used to find the dot product of two vectors. It calculates the dot product of** 2-dimensional** & **3-dimensional **vectors. This calculator provides a step-by-step solution to the given problems.

## What is a dot product?

The dot product is a fundamental way of combining two vectors. It is used to know the direction of two vectors. The **dot product** of two vectors is also known as the resultant and it is a **scalar quantity**.

The dot product is denoted by **a.b** and reads as “`a dot b`

”.

## Formula of the dot product

The formula of the dot product is:

**a.b = |a||b| cos(θ) … (i)**

**a & b**are two vectors.**|a| & |b|**are the magnitudes of vectors**a & b**.- “
**θ**” is the angle between**a & b**.

In mathematics, we can derive another form of the dot product. Let **a = (a _{1}, a_{2}, a_{3})** &

**b = (b**be two vectors. Now according to the Law of cosines,

_{1}, b_{2}, b_{3})`|a – b|`

^{2} = |a|^{2} + |b|^{2} – 2|a||b| cos(θ)

Let’s simplify the above expression with the help of the magnitude of a vector formula.

`(a`

_{1} – b1)^{2 }+ (a_{2} – b_{2})^{2} + (a_{3} – b_{3})^{2} = (a_{1}^{2} + a_{2}^{2 }+ a_{3}^{2}) + (b_{1}^{2} + b_{2}^{2} + b_{3}^{2}) – 2|a||b| cos(θ)

Open the square of each term.

`(a`

_{1}^{2} - 2a_{1}b_{1} + b_{1}^{2}) + (a_{2}^{2} - 2a_{2}b_{2} + b_{2}^{2}) + (a_{3}^{2} - 2a_{3}b_{3} + b_{3}^{2}) = (a_{1}^{2} + a_{2}^{2 }+ a_{3}^{2}) + (b_{1}^{2} + b_{2}^{2} + b_{3}^{2}) – 2|a||b| cos(θ)

Now take the similar terms on the same side of the equation.

`(a`

_{1}^{2} - 2a_{1}b_{1} + b_{1}^{2}) + (a_{2}^{2} - 2a_{2}b_{2} + b_{2}^{2}) + (a_{3}^{2} - 2a_{3}b_{3} + b_{3}^{2}) - (a_{1}^{2} + a_{2}^{2 }+ a_{3}^{2}) - (b_{1}^{2} + b_{2}^{2} + b_{3}^{2}) = – 2|a||b| cos(θ)

After subtracting the similar terms, we get

- 2a_{1}b_{1} + (-2a_{2}b_{2}) + (-2a_{3}b_{3}) = – 2|a||b| cos(θ)

- 2(a_{1}b_{1} + a_{2}b_{2 }+ a_{3}b_{3}) = – 2|a||b| cos(θ)

`(a`

_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}) = |a||b| cos(θ) … (ii)

Now put equation **(ii) **in **(i)**, we get

**a.b = (a _{1}b_{1} + a_{2}b_{2} + a_{3}b_{3})**

This formula is used to find the dot product of two vectors.

## How to calculate the dot product of two vectors?

Follow the below example to learn how to calculate the dot product of two vectors.

**Example**

Find the dot product of vectors **a = (2i, 3j, 6k)** and **b = (4i, 7j, 9k)**.

**Solution**

**Step 1: **Take the given vectors.

a = (2i, 3j, 6k)

b = (4i, 7j, 9k)

**Step 2:** Take the formula of the dot product.

`a.b = (a`

_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3})

**Step 3: **Substitute the given values of vectors **a & b** in the formula.

a.b = ((2)(4) + (3)(7) + (6)(9))

a.b = (8 + 21 + 54)

`a.b = 83 `