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

## What is 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 read 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 dot product. Let a = (a_{1}, a_{2}, a_{3}) & b = (b_{1}, b_{2}, b_{3}) be two vectors. Now according to the Law of cosines,

|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 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 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