# What is the dot product of two vectors?

Dec 25, 2014

The dot product of two vectors is a quite interesting operation because it gives, as a result, a...SCALAR (a number without vectorial properties)!

As a definition you have:

Given two vectors $\vec{v}$ and $\vec{w}$ the dot product is given by:

$\vec{v} \cdot \vec{w} = | \vec{v} | \cdot | \vec{w} | \cdot \cos \left(\theta\right)$

i.e. is equal to the product of the modules of the two vectors times de cosine of the angle between them.

For example:
if $| \vec{v} | = 10$ and $| \vec{w} | = 5$ and theta =60°
vecv*vecw=|vecv|*|vecw|*cos(theta)=10*5*cos(60°)=25

Another way of calculating the dot product is to use the coordinates of the vectors:
If you have:
$\vec{v} = a \vec{i} + b \vec{j} + c \vec{k}$ and $\vec{w} = \mathrm{dv} e c i + e \vec{j} + f \vec{k}$
(where $a , b , c , d , e \mathmr{and} f$ are real numbers)
you can write:
$\vec{v} \cdot \vec{w} = \left(a \cdot d\right) + \left(b \cdot e\right) + \left(c \cdot f\right)$
For example:
if:
$\vec{v} = 3 \vec{i} + 5 \vec{j} - 3 \vec{k}$ and $\vec{w} = - 1 \vec{i} + 2 \vec{j} + 3 \vec{k}$
$\vec{v} \cdot \vec{w} = \left(3 \cdot - 1\right) + \left(5 \cdot 2\right) + \left(- 3 \cdot 3\right) =$
$= - 3 + 10 - 9 = - 2$

This operation has important practical applications. For example in Physics the dot product of Force (a vector) and displacement (a vector) gives as a result a number without vectorial characteristics, called, Work.