# Dot Product of Two Vectors - Calculator

An online calculator to calculate the dot product of two vectors also called the scalar product.

## Use of Dot Product Calculator

1 - Enter the components of the two vectors as real numbers in decimal form such as 2, 1.5, ... and press "Calculate the dot Product". The answer is a scalar.

Characters other than numbers are not accepted by the calculator.

## Definition of the Dot Product of two Vectors

Let u and v be two 3D vectors given in component form by

u = < a , b, c > and v = < d , e , f >

The dot product of the two vectors u and v above is given by

u.v = < a, b , c > ? < d , e , f > = a×d + b×e +c×f

and it is a scalar quantity.

Example 1

Let u = < -2 , 3, 2 > and v = < 0 , -1 , 6 >

The dot product of the vector u and v is given by

u ? v = < -2 , 3, 2 > ? < 0 , -1 , 6 > = (-2)×(0) + 3×(-1) + 2×6 = 9

## Applications of the Dot Product

The dot product has many applications in mathematics, physics, engineering, ... We will give some examples below.

Example 2

The dot product can be used to find out if two vectors are orthogonal (i.e they are perpendicular or their directions make 90 degrees).

The geometric definition of the dot product is

u ? v = || u || || v || cos (?)

where ? is the angle between vectors u and v.

Hence, the dot product of two orthogonal vectors is equal to zero since cos(90°) = 0.

As an example, let u = < 3 , 3, 3 > and v = < - 2 , 2 , 0 >

The dot product of the vector u and v is given by

u.v = < 3 , 3, 1 > ? < - 2 , 2 , 0 > = 3×(-2) + 3×(2) + (3)×(0) = 0

Conclusion: vector u < 3 , 3, 3 > and v < - 2 , 2, 0 > are orthogonal; see the vector plot in the figure below.

Example 3

The dot product may be used to find the angle ? between two vectors given by their components in 2 or 3 dimensional spaces.

cos(?) = u ? v / || u || || v ||

As an example, let u = < 4 , 3, 0 > and v = < 0 , 8 , 6 >

Calculate the dot product using the components.

u ? v = (4)(0) + (3)(8) + (0)(6) = 24

Calculate the magnitudes || u || and || v ||

|| u || = √(4^{ 2} + 3^{ 2} + 0^{ 2}) = 5

|| v || = √(0^{ 2} + 8^{ 2} + 6^{ 2}) = 10

cos(?) = u ? v / || u || || v || = 24 / (5×10)

? = arccos(24/50) = 61.3°

Example 4

In physics, the work W done by a constant force F acting on an object along a constant direction for a displacement d is given by

W = F ? d

## More References and Links

Dot Product of Two Vectors and Applications.

3D Vectors.

vectors.