# How to Find Dot / Scalar Product of Vectors - Tutorial

## How to Find Dot Product of Vectors - Tutorial, Formula, Example

##### Definition:

The dot product (also called the inner product or scalar product) of two vectors is defined as:

#### Formula :

→ → a . b  = │ a │.│ b │ cos θ Where, |A| and |B| represents the magnitudes of vectors A and B theta is the angle between vectors A and B.

#### Dot product calculation :

The dot or scalar product of vectors A = a1 i + a2 j and B = b1 i + b2 j can be written as A . B = a1 . b1 + a2 . b2
##### Example (calculation in two dimensions):

Vectors A and B are given by A = 5i + 2j and B = 3i + 4j. Find the dot product of the two vectors.

##### Solution:

A . B = (5i + 2j ) . ( 3i + 4j ) A . B = 5 . 3 + 2 . 4 A . B = 23

##### Example (calculation in three dimensions):

Vectors A and B are given by A = 4i + 2j + 1kand B = 5i + 4j+ 2k. Find the dot product of the two vectors.

##### Solution:

A . B = (4i + 2j+ 1k ) . ( 5i + 4j+ 2k ) A . B = 4 . 5 + 2 . 4 + 1 . 2 A . B = 30

##### Example ( The angle between two vectors : )

Vectors A and B are given by A = 5i + 4j - 3kand B = -3i + 2j- 1k. Find the angle between two vectors A and B.

##### Solution:

A . B = ( 5i + 4j - 3k ) . ( -3i + 2j- 1k ) a . b = 5.(-3) + 4.(2) + (-3).(-1) = -15+8+3 = -4 |a| = √(52 + 42 + (-3)2) = √50 |b| = √(-32 + (2)2 + (-1)2) = √14 a . b  = │ a │.│ b │ cos θ cos θ = -4 / √50 √14 θ = cos-1(-0.151185789) θ = 98.695651011