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# Dot Product Calculator

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
 1 What is the Dot Product? 2 What is Dot Product Formula? 3 How to Do Dot Product Manually (Step-by-Step): 4 Applications of Dot Product: 5 Why is dot product scalar? 6 What is the dot product of i and j? 7 What does it mean when the dot product is 0? 8 What does it mean when the dot product is 1? 9 Can dot product be negative?
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An online vector dot product calculator allows you to find the resultant of the two vectors by multiplying with each other. This online 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.

Keep reading to know more about the vector multiplication problems, manual calculations, definition & dot product formula and much more. Let’s start with the basics.

Also, you can use our online cross product calculator that helps you to find the cross product between the two vector with complete manual calculation.

## 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 online dot product calculator also 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 (Step-by-Step):

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

### Calculation With Magnitude & Angle:

From these input parameters, we have to know the magnitude of both vectors & the angle between the vectors. Here we have an example:

Example:

A vector has magnitude 10 and b vector has magnitude of 15, the angle between the vectors is 60degree. Find the dot product of two vectors?

Solution:

Step 1:

Here,

|a| = 10

|b| = 15

Θ = 60degree

Step 2:

a.b = |a| |b| cosΘ

a.b = 10 * 15 cos60

a.b = 150 cos60

a.b = 150 (0.5)

a.b = 75

## Applications of Dot Product:

There are many uses & applications of the scalar product, here we mentioned some of these:

• It is helpful in determining whether two vectors are perpendicular or parallel to each other.
• The law of cosine can be proved with the help of dot product. Cosine law is as follows:= a2 + b2 – 2abcosΘ
• Many other physical quantities defined as a dot product. For example:
• Power
• Electric/magnetic flux
• Work
• Magnetic potential energy

## How to Use this Online Dot Product Calculator:

The calculations become very easy with this online free calculator. This tool determines 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.

Note:

No matter, which method or input parameters you selected, the online dot product calculator shows you the accurate result according to your selected option.

### Why is dot product scalar?

It’s other name is scalar product because it is a scalar quantity while the cross product is a vector. The dot product is defining the components of one vector in the direction of the other, when the second vector is normalized. Such that it is a scalar multiplier.

### What is the dot product of i and j?

The scalar product between i, j & k with each other always gives zero result while it gives 1 if they multiply with itself.

### What does it mean when the dot product is 0?

When two vectors are mutually perpendicular to each other, it means the angle between vectors is 90degree, then the scalar product is 0.

### What does it mean when the dot product is 1?

When two vectors are parallel to each other or moving in the same direction, it means the angle between vectors is 0degree, then the scalar product is 1.

### Can dot product be negative?

If this happens, it means the angle between vectors is greater than 90degree & one vector has a component in the opposite direction of the other.

## Final Words:

As the dot product has a wide range of applications in physics, geometry and other fields. It is helpful to test whether or not the two vectors are orthogonal. So, simply try this online vector dot product calculator that helps you to find the dot product of two vectors. This tool specifically designed for students and professionals to solve their physics-related problems.

## References:

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

From the site of Math Insight : Formula for the dot-product in terms of vector component.

From the source of khan Academy : To compute the dot product.