# Dot Product Calculator

Calculate the dot product of two vectors using the calculator below. See the steps to solve with the solution below.

## Dot Product of Vectors (a · b):

### Steps to Solve

#### Use the Dot Product Formula

a·b = (x_{a} · x_{b}) + (y_{a} · y_{b})

#### Substitute Values and Solve

Enter vectors a & b above to see the solution here

### Steps to Solve

#### Use the Dot Product Formula

a·b = (x_{a} · x_{b}) + (y_{a} · y_{b}) + (z_{a} · z_{b})

#### Substitute Values and Solve

Enter vectors a & b above to see the solution here

## On this page:

## How to Calculate the Dot Product of Two Vectors

When working with vectors, a dot product is the sum of the products of each component in the cartesian coordinates of two vectors, *a* and *b*. Unlike the cross product, a dot product is a single number rather than a vector and is denoted *a·b*.

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

### Dot Product Formula

The dot product formula is given:

a·b = |a|·|b|·cos(θ)

Where:

*|a|*= magnitude of vector*a**|b|*= magnitude of vector*b**θ*= angle between the vectors

You can use our magnitude and angle between two vectors calculators to solve for *|a|*, *|b|*, and *θ*.

#### Practical Application

You can use an alternative formula to reduce the complexity of calculating the dot product by multiplying the corresponding components of each vector’s coordinate.

a·b = (x_{a} · x_{b}) + (y_{a} · y_{b}) + (z_{a} · z_{b})

To use the formula, substitute the values of two vectors for x_{a}, y_{a}, z_{a}, x_{b}, y_{b}, & z_{b} to solve the dot product.

To solve it, substitute the values for each vector and solve.

**For example,** let’s find the dot product of the vectors (1, 7, 3) and (4, 2, 1).

Start by substituting the values in the formula above.

a·b = (1 · 4) + (7 · 2) + (3 · 1)

Then solve.

a·b = 4 + 14 + 3

a·b = 21

You might also be interested in our vector addition and vector subtraction calculators.