Question

What is the dot product of two vectors?

Answer 1

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 `vecv` and `vecw` the dot product is given by:

`vecv*vecw=|vecv|*|vecw|*cos(theta)`

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 `|vecv|=10` and `|vecw|=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:
`vecv=aveci+bvecj+cveck` and `vecw=dveci+evecj+fveck`
(where `a,b,c,d,e and f` are real numbers)
you can write:
`vecv*vecw=(a*d)+(b*e)+(c*f)`
For example:
if:
`vecv=3veci+5vecj-3veck` and `vecw=-1veci+2vecj+3veck`
`vecv*vecw=(3*-1)+(5*2)+(-3*3)=`
`=-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.