Question

How do you use the definition of the scalar product, find the angles between the following pairs of vectors: - 4i + 5 j- k and 3i + 4j - k?

Answer 1

The angle is `=74.2^@`

Explanation:

The vectors are

`vecA= <-4,5,-1>`

and

`vecB= <3,4,-1>`

The angle between `vecA` and `vecB` is given by the dot product definition.

`vecA.vecB=∥vecA∥*∥vecB∥costheta`

Where `theta` is the angle between `vecA` and `vecB`

The dot product is

`vecA.vecB=〈-4,5,-1〉.〈3,4,-1〉=-12+20+1=9`

The modulus of `vecA`= `∥〈-4,5,-1〉∥=sqrt(16+25+1)=sqrt42`

The modulus of `vecB`= `∥〈3,4,-1〉∥=sqrt(9+16+1)=sqrt26`

So,

`costheta=(vecA.vecB)/(∥vecA∥*∥vecB∥)=9/(sqrt42*sqrt26)=0.27`

`theta=arccos(0.27)=74.2^@`