## Introduction to Dot Product Calculator:

Dot product calculator is an online tool that helps you to **find the dot product** of given vectors in a fraction of seconds. It is used to determine the product of two or more vectors after multiplying it with their crosspounding elements.

Our dot product of vectors calculator is an educational tool that helps teachers, educators, and researchers to easily get the solution of dot product problems without doing manual calculations quickly.

## What is a Dot Product?

Dot product is an algebraic operation in which **vector cross-pounding** elements are multiplied. It is known as a scalar product or inner product. It simplifies the vector quantities into a scalar quantity.

It is mostly used in mathematics and physics where the dot product is used to find the angle between two vector quantities, its projection or direction, etc.

## Formula of the Dot Product:

The **dot product formula** consists of vector or scalar quantities. The vector dot product calculator uses the following formula in which a and b are two scalar quantities while ai and bi are the vector quantities.

$$ a . b \;=\; \sum_{i=1}^{n} a_i b_j $$

## Rules of the Dot Product:

For the calculation of dot product problems, the inner product calculator uses some basic **rules** that help to evaluate the multiplication of the vector without any trouble. These rules are:

**Commutative Law**:

The dot product is commutative, which means that the order of the vectors in the dot product does not matter as,

$$ A . B \;=\; B . A $$

**Distributive Law**:

The distributive property of addition holds for the dot product of vectors:

$$ A . (B + C) \;=\; A . B + A . C $$

**Scalar Multiplication Property**:

The dot product of a vector can be multiplied with a scalar vector because it has no dot product of the vectors change into scaler in solution:

$$ (cA) . B \;=\; c(A . B) $$

Where c is a scalar.

**Zero Vector**:

When you take the dot product of any vector with the zero vector, its result will be zero.

$$ A . 0 \;=\; 0 $$

Orthogonality or Perpendicular Property:

If the dot product of two vectors is equal to zero, that means they are perpendicular to each other.

$$ if \; A . B \;=\; 0 $$

$$ Then\; A \bot B $$

**Magnitude of Vector**:

When you take the dot product of a vector with itself vector, it gives the square of magnitude.

$$ A . A \;=\; \Vert A \Vert^2 $$

**Angle Between Vectors**:

The dot product can be used to find the angle between two vectors, whereas θ is the angle between 𝐴 and 𝐵

$$ A . B \;=\; \Vert A \Vert \Vert B \Vert cos\; \theta $$

## How to Calculate the Dot Product?

To **calculate the dot product** of two vectors, let's suppose an example that helps you to understand the calculation process of the dot product in steps.

Let's calculate the dot product of two 3-dimensional vectors A and B as A = [2, 3, 4], B = [1, 0, −1]:

**Step 1**:

The dot product calculator identifies whether both vectors have the same number of components or not. If they have the same length of component then you can use it to calculate the dot product as the given example has A = [2, 3, 4], B = [1, 0, −1].

$$ A \;=\; [2, 3, 4],\; B \;=\; [1,0, -1] $$

**Step 2**:

Then the vector calculator dot product multiplies each component of the first vector by its corresponding component of the second vector as:

$$ A . B \;=\; (2.1, 3.0, 4.-1) $$

**Step 3**:

The dot product of two vectors calculator **adds all the elements** obtained after the dot product of the given vectors.

$$ A . B \;=\; 2 + 0 + (-4) $$

$$ A . B \;=\; 2 - 4 $$

$$ A . B \;=\; -2 $$

So, the result of the dot product,

$$ A ⋅ B \;is\; −2 $$

The dot product vector calculator uses these steps for calculation which you can use as well to calculate the dot product for any two vectors that have an equal length.

## Solved Example of the Dot Product:

Let us see practical **examples of the dot product** of two vector problems that explain the calculation process of the vector dot product calculator.

### Example: Find the Dot Product of u =〈3, 5, 2〉and v = 〈-1, 3, 0〉

**Solution**:

$$ u . v \;=\; u1 v1 + u2v2 + u3v3 $$

$$ =\; 3(-1) + 5(3) + 2(0) $$

$$ =\; -3 + 15 + 0 $$

$$ =\; 12 $$

**Example**:

Find the dot product of the following:

Let a = 〈1, 2, -3〉, b = 〈0, 2, 4〉and c = 〈5, -1, 3〉

**Solution**:

$$ (a . b) c \;=\; (\langle 1, 2, -3 \rangle . \langle 0, 2, 4 \rangle) \langle 5, -1, 3 \rangle $$

$$ =\; (1(0) + 2(2) + (-3)(4)) \langle 5, -1, 3 \rangle $$

$$ =\; -8 \langle 5, -1, 3 \rangle $$

$$ =\; \langle -40, 8, -24 \rangle $$

## How to Use Dot Product Calculator?

The dot product of vectors calculator has a simple layout that helps you to solve the dot product of vectors. You just need to put your vector value in this calculator and the result will appear immediately without any delay. Follow some instructions to know how it is used.

- Enter the value of vector A in the input field of the inner product calculator.
- Enter the value of vector B in the input field
- Check your given input vector value before clicking the calculate button to get the result of the dot product after calculation.
- Click the “
**Calculate**” button for the solution for dot product problems. - Click the “Recalculate” button for the evaluation of more examples of the dot product problem with a solution.

## Output of Dot Product of Vectors Calculator:

Dot Product Calculator provides you with a solution of dot product for vectors as per your input when you click on the calculate button. It may include as:

**In the Result Box**

When you click on the result button you get the **solution** to the scalar product problem.

**Possible Steps Box**

Click on the steps option of the vector calculator dot product so that you get the solution of vectors after taking their dot product in a step-by-step method.

## Benefits of Inner Product Calculator:

The dot product of two vectors calculator has serval benefits whenever you use it to solve dot product questions and get their solution. It keeps you away from a manual calculator that just takes the input value and gives a solution without imposing a condition of a sign-up option.

- The dot product vector calculator is a
**trustworthy tool**as it always provides you with accurate solutions for scalar products of vector questions. - It is a speedy tool that evaluates dot products problems with solutions in run of time
- The dot product of vectors calculator is a learning tool that helps children to get a better understanding of dot product concepts without going to any tutor.
- It is a handy tool that solves dot product problems for different dimensions and you do not need to put in external effort.
- The vector calculator dot product is a free tool that allows you to use it for the calculation of dot products without taking any fee.
- It is an easy-to-use tool, anyone or even a beginner can use it for the solution of the dot product of vectors.
- Vector dot product calculator can operate on a desktop, mobile, or laptop through the internet and get solutions of dot product problems.