Dot Product Calculator

Dot product calculator

The online vector dot product calculator allows you to find the total dot product of two vectors by multiplying the two vectors with each other.

This online calculator will help you compute:

  • Vector components



  • Magnitude and angle

a · b =

0 * 0 * cos(0) = 0

Magnitude of a =

Magnitude of b =

Angle θ between vector a and b =


The resultant of this calculation i.e. the total dot product is a scalar, not a vector. The dot product finds the total length of the two vectors (length). The i, j, and k fields are multiplied together, and then all values are added up to give the total dot product.

How to use the dot product calculator

The dot product calculator, also known as the dot product of two vectors calculator or matrix dot product calculator, is straightforward to use. Simply enter the required values and use our online calculator to find the total dot product in a few easy steps:

  1. First, input the 3 values for vector a (x, y, z).
  2. Second, input the 3 values for vector b (x, y, z).
  3. Finally, after you've entered all the values, click calculate. You'll automatically get the solution, the dot product of vector c (x, y, z) and the angle between vectors.

Note: You must input at least 2 vector values and the number of terms must be the same for all vectors. You can enter the values in parentheses, square brackets, greater or less than signs or simply begin a new line for each vector. Values entered may be decimals or integers, but cannot be fractions or functions.

What is a dot product?

In linear algebra, a dot product (also known as a scalar product or inner product) is a result of multiplying the numerical values of two or more vectors. The scalar product of two vectors is equal to the product of their magnitudes. Therefore, two perpendicular vectors will have a dot product of zero.

Dot product formula

For vectors a = (a1, a2, a3) and b = (b1, b2, b3)the dot product can be found by using the following formula:

a · b = a1b1 + a2b2 + a3b3

Given that the dot product is the product of the magnitudes of vectors multiplied by the value of the cosine of the angle between the vectors, this can be expressed as the following equation: 

ab = a b cosθ

So, if a vector had 3 components, the dot product formula would be:

a•b = a₁ * b₁ + a₂ * b₂ + a₃ * b₃

Since cosine is the ratio of the vectors’ magnitudes and the scalar products. This can be further expressed as:

cosα = a•b / (|a| * |b|).


  • a and b are two vectors
  • |a| and |b| are the magnitudes of the vector a and b respectively
  • Θ is the angle between the two vectors

The dot product calculator can also be used to solve for the angle between two given vectors. To find the angle Θ between the vectors, use the following equation:

Θ = Cos-1 a.b / |a| |b|

How to calculate the dot product manually

To perform the dot product calculator yourself by hand, you'll need to draw both vectors a and b and separate them with an angle. Note this only works well for 2D vectors.

Alternatively, using the formulas we've provided above, you should now be able to solve for the dot product or angle between two given vectors when given input variables.

Solving dot product with vector component - Example 1

Let's assume we know the input variables for the two vectors a and b.

  • Example: If vector a = [4, 5, -3] and vector b = [1, -2, -2]. How can we find out the dot product of these two vectors?
  • Input the values into the equations.
  • a ⋅ b = (a1 * b1) + (a2 * b2) + (a3 * b3)
  • a ⋅ b = (4 * 1) + (5 * -2) + (-3 * -2)
  • a ⋅ b = 4 -10 +6
  • a ⋅ b = 0

Solving dot product with magnitude and angle - Example 2

In order to solve for the dot product in this example, we need to know the magnitude for both vectors and the angle between the two vectors.

  • 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?


Step 1:


|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


What is the dot product of a vector with itself?

The dot product of a vector with itself is the square of its magnitude. Since the dot product of two vectors is commutative, the order of the vectors in the product does not matter.

Applications of the dot product

Some applications of the dot product include:

  • determining whether two vectors are perpendicular or parallel to each other
  • proving the law of cosine
  • physical quantities defined as a dot product such as power, electric or magnetic flux and magnetic potential energy