# Dot Product Calculator

Dot Product Calculator computes the dot product of the two given vectors. When two vectors are multiplied using dot product then the quantity so obtained will be a scalar. The dot product could be a positive or negative real number.

## What is Dot Product Calculator?

Dot Product Calculator is an online tool that helps to determine the scalar quantity that is a result of the dot product of the given two vectors. The quantity that results after taking the dot product will be in the same plane as the two given vectors. To use the dot product calculator, enter the values in the given input boxes.

### Dot Product Calculator

## How to Use Dot Product Calculator?

Please follow the below steps to calculate the dot product of the two given vectors using the dot product calculator

**Step 1:**Go to Cuemath's online dot product calculator.**Step 2:**Enter the coefficients of two vectors in the given input boxes.**Step 3:**Click on the**"Multiply"**button to calculate the dot product.**Step 4:**Click on the**"Reset"**button to clear the fields and enter the new values.

## How to Find Dot Product Calculator?

The dot product is defined as the product of the magnitude of the two vectors and the cosine of the angle between the two given vectors. If \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are two vectors then the dot product is given as follows:

\(\overrightarrow{a}\). \(\overrightarrow{b}\) = |\(\overrightarrow{a}\)|.|\(\overrightarrow{b}\)|cosθ.

Suppose we are given two vectors that are expressed in the form of their unit vectors i, j, k along the x, y, and z-axis. Then the steps that are followed to find the dot product between the two vectors are given below:

\(\overrightarrow{a}\) = \(a_{1}\hat{i} + a_{2}\hat{j} + a_{3}\hat{k}\)

\(\overrightarrow{b}\) = \(b_{1}\hat{i} + b_{2}\hat{j} + b_{3}\hat{k}\)

To find the dot product

\(\overrightarrow{a}\). \(\overrightarrow{b}\) = (\(a_{1}\hat{i} + a_{2}\hat{j} + a_{3}\hat{k}\)).(\(b_{1}\hat{i} + b_{2}\hat{j} + b_{3}\hat{k}\))

= \((a_{1}b_{1})(\hat{i}.\hat{i}) + (a_{1}b_{2})(\hat{i}.\hat{j}) + (a_{1}b_{3})(\hat{i}.\hat{k})\) + \((a_{2}b_{1})(\hat{j}.\hat{i}) + (a_{2}b_{2})(\hat{j}.\hat{j}) + (a_{2}b_{3})(\hat{j}.\hat{k})\) + \((a_{3}b_{1})(\hat{k}.\hat{i}) + (a_{3}b_{2})(\hat{k}.\hat{j}) + (a_{3}b_{3})(\hat{k}.\hat{k})\)

\(\hat{i}.\hat{j}\) = \(\hat{i}.\hat{k}\) = \(\hat{k}.\hat{j}\) = cos 90 = 0. This is because these vectors are orthogonal.

\(\hat{i}.\hat{i}\) = \(\hat{k}.\hat{k}\) = \(\hat{j}.\hat{j}\) = cos 0 = 1. Because these vectors are codirectional.

\(\overrightarrow{a}\). \(\overrightarrow{b}\) = \(a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}\)

## Solved Examples on Dot Product Calculator

**Example 1:**

Find the dot product of two vectors \(\overrightarrow{a}\) = \(4\hat{i} + 2\hat{j} - 5\hat{k}\) and \(\overrightarrow{b}\) = \(3\hat{i} - 2\hat{j} - \hat{k}\). Verify the result using the dot product calculator.

**Solution:**

Given \(\overrightarrow{a}\) = \(4\hat{i} + 2\hat{j} - 5\hat{k}\) and \(\overrightarrow{b}\) = \(3\hat{i} - 2\hat{j} - \hat{k}\)

\(\overrightarrow{a}\). \(\overrightarrow{b}\) = \(a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}\)

\(\overrightarrow{a}\). \(\overrightarrow{b}\) = (4 . 3) + (2 . (-2)) + ((-5) . 1)

= 12 - 4 - 5

= 3

Therefore, the dot product of two vectors is 3.

**Example 2:**

Find the dot product of two vectors \(\overrightarrow{a}\) = \(2.3\hat{i} - 1.2\hat{j} + 8.9\hat{k}\) and \(\overrightarrow{b}\) = \(-4.6\hat{i} + 2.8\hat{j} + 5.5\hat{k}\). Verify the result using the dot product calculator.

**Solution:**

Given \(\overrightarrow{a}\) = \(2.3\hat{i} - 1.2\hat{j} + 8.9\hat{k}\) and \(\overrightarrow{b}\) = (-4.6\hat{i} + 2.8\hat{j} + 5.5\hat{k}\)

\(\overrightarrow{a}\). \(\overrightarrow{b}\) = \(a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}\)

\(\overrightarrow{a}\). \(\overrightarrow{b}\) = (2.3 . 3) + (2 . (-2)) + ((-5) . 1)

= 12 - 4 - 5

= 3

Therefore, the dot product of two vectors is 3.

Similarly, you can use the calculator to find the dot product of two vectors for the following:

- a = 4i + 2j - 5k and b = -1i + 4j - 3k
- a = -2i - 5k and b = -7i + j + k

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