Dot Product Calculator

Calculate the dot product of two vectors quickly and accurately. Enter your vector values and get the scalar product with a step-by-step multiplication and addition breakdown.

Calculate Vector Dot Product

Enter the values for both vectors and calculate instantly.

Select Vector Dimension

Vector A

Vector B

Dot Product Result

What Is a Dot Product?

The dot product, also known as the scalar product, is a mathematical operation performed on two vectors with the same number of components. The result of a dot product is a single numerical value rather than another vector.

To calculate the dot product, multiply each corresponding component of the two vectors and then add all of the resulting products together. This calculator performs the entire calculation automatically and provides a clear breakdown of the formula.

A · B = A₁B₁ + A₂B₂ + A₃B₃ + ... + AₙBₙ

For example, if Vector A is (2, 3, 4) and Vector B is (5, 6, 7), the dot product is calculated by multiplying corresponding values: (2 × 5) + (3 × 6) + (4 × 7). The sum of these values gives the final scalar result.

How Does the Dot Product Calculator Work?

This calculator uses the standard vector dot product formula. Both vectors must have the same number of dimensions because each value in the first vector needs a corresponding value in the second.

Select the number of dimensions for your vectors.
Enter each numerical component of Vector A.
Enter the corresponding numerical components of Vector B.
Multiply each pair of corresponding values together.
Add all multiplication results to get the final dot product.

Dot Product Formula

Suppose two vectors are represented as:

A = (a₁, a₂, a₃, ..., aₙ)
B = (b₁, b₂, b₃, ..., bₙ)

Their dot product is calculated as:

A · B = a₁b₁ + a₂b₂ + a₃b₃ + ... + aₙbₙ

The result is called a scalar because it has magnitude but does not have a direction. Dot products are commonly used in mathematics, physics, engineering, computer graphics, machine learning, and many other technical fields.

Example of a Dot Product Calculation

Consider the following two vectors:

A = (3, 4, 2)
B = (5, 1, 6)

First, multiply the corresponding components:

(3 × 5) + (4 × 1) + (2 × 6)

Then calculate the products:

15 + 4 + 12 = 31

Therefore, the dot product of Vector A and Vector B is 31.

Where Is the Dot Product Used?

Dot products are used in a wide range of mathematical and practical applications. In geometry, they can help determine the angle between two vectors. A dot product of zero can indicate that two non-zero vectors are perpendicular to each other.

In physics and engineering, scalar products are used when calculating concepts involving direction and magnitude. Computer graphics also uses dot products for lighting calculations, surface orientation, reflections, and three-dimensional rendering.

In data science and machine learning, vectors are frequently used to represent numerical data. Dot products can therefore play an important role in similarity calculations, mathematical models, optimization methods, and neural network operations.

Frequently Asked Questions

The dot product is a scalar value obtained by multiplying corresponding components of two vectors and adding all of the products together.

No. For a standard dot product calculation, both vectors must contain the same number of components.

If the dot product of two non-zero vectors is zero, the vectors are perpendicular or orthogonal to each other.

No. A dot product can be positive, negative, or zero depending on the values and directions of the vectors.

The dot product is also commonly called the scalar product or inner product in many mathematical contexts.

Yes. You can select dimensions from 2 through 10 and calculate the dot product for vectors with the same number of components.