Dot Product Calculator

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

Vector a
Vector b
Vector a
Vector b

Dot Product of Vectors (a · b):


Steps to Solve

Use the Dot Product Formula

a·b = (xa · xb) + (ya · yb) + (za · zb)

Substitute Values and Solve

Enter vectors a & b above to see the solution here

Learn how we calculated this below

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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)


  • |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 = (xa · xb) + (ya · yb) + (za · zb)

To use the formula, substitute the values of two vectors for xa, ya, za, xb, yb, & zb 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.