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Or # Dot Product Calculator

This calculator will either take coordinates (2D or 3D) or angle along with magnitude to determine the dot product of two vectors, with the steps shown.

Vector(A) Representation

First vector (a)

$$\vec i$$
$$\vec j$$
$$\vec k$$

Initial Point (A)

Terminal Point (B)

Vector(B) Representation

Second vector (b)

$$\vec i$$
$$\vec j$$
$$\vec k$$

Initial Point (A)

Terminal Point (B)

|A|

|B|

θ (deg)

Table of Content

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Dot product calculator is free tool to find the resultant of the two vectors by multiplying with each other. This calculator for dot product of two vectors helps to do the calculations with:

• Vector Components, it can either be 2D or 3D vector.
• Magnitude & angle.

When it comes to components, you can be able to perform calculations by:

• Coordinates.
• Points.

## What is the Dot Product?

It is also known as scalar product, and can be defined as “the sum of the component wise products”. The scalar product of two vectors is equal to the product of their magnitudes.

So, it is zero for two mutually perpendicular vectors and it is denoted with a symbol ‘.’ between the two vectors.

The main difference between dot & cross product is that the product of dot operation is a single number while the outcome of cross operation is a vector.

### What is Dot Product Formula?

The formula for the scalar multiplication between two vectors is as follow:

a.b = |a| |b| cosΘ

Where,

a & b are two vectors while |a| & |b| are the magnitudes of the vector a & b respectively.
Θ is the angle between the two vectors.

Our calculator works to find the angle Θ between the vectors by using the following equation:

Θ = Cos-1 a.b / |a| |b|

## How to Do Dot Product Manually:

The formula for the calculations is discussed above, now we have manual examples for both the methods.

### Calculation With Vector Component:

From these input parameters, we have to know the two coordinates for which we are going to do calculations. Here we have an example:

Example:

If the vector a = [2,-4,3] & second vector b = [-4,3,5]. What is dot product of two vectors?

Solution:

Step 1:

Find the product of the first component of each vector.

So, (2)*(-4) = -8

Step 2:

Find the product of the second component of each vector.

So, (-4)*(3) = -12

Step 3:

Find the product of the third component of each vector.

So, (3)*(5) = 15

Step 4:

Add all these values to find the scalar product (dot product)So, (2)*(-4) = -8

(-8)+(-12)+15

-8 – 12 + 15

a.b = -5

If we have to find the angle between the two vectors, then use the formula as:

Θ = Cos-1 a.b / |a| |b|

Step 1:

Magnitude of vector a.

|a| = √ (2)2 + (-4)2 + (3)2

|a| = √ 4+ 16 + 9

|a| = √ 29

|a| = 5.38

Step 2:

Magnitude of vector b.

|b| = √ (-4)2 + (3)2 + (5)2

|b| = √ 16+ 9 + 25

|b| = √ 50

|b| = 7.07

Step 3:

Θ = Cos-1 a.b / |a| |b|

Θ = Cos-1 -5 / 5.38* 7.07

Θ = Cos-1 -5 / 38.03

Θ = Cos-1 -0.1314

Θ = 97.53deg

## How to Use Our Calculator:

The dot product of vectors with two different methods that we are going to discuss:

### Vector Components:

For calculations from this method, just stick to the following points:

Inputs:

• First of all, select the dimension from the tab. It is either 2D or 3D.
• Very next, choose the vector representation for the first vector from the dropdown of the calculator.
• Then, choose the vector representation for the second vector from the dropdown of this tool.
• Enter in all the fields, according to the selected option.
• Lastly, hit the calculate button.

### Magnitude & Angle:

For calculations from this method, just follow the following points:

Inputs:

• First of all, enter the magnitude of the first vector.
• Very next, enter the magnitude of the second vector in the designated field.
• After this, plug-in the angle between the vectors.
• Finally, click on the calculate button.

Outputs:

Once you enter in all the fields, the calculator shows:

• Dot product between vectors.
• Magnitude of vector A.
• Magnitude of vector B.
• Angle between the vectors.
• Step-by-Step calculations.

## References:

From the authorized source of Wikipedia: Definition & properties of dot product.