Question

How do you use the definition of the scalar product, find the angles between the following pairs of vectors: A = -i - 2k and B = - 5i + 5j + 5k?

Answer 1

The angle is `=105º`

Explanation:

The dot product of `vecA` and `vecB`
is given by `vecA.vecB=∥vecA∥.∥vecB∥costheta`
where `theta` is the angle between the 2 vectors.
Here, `vecA=〈-1,0,-2〉` and `vecB=〈-5,5,5〉`

The dot product `vecA.vecB=〈-1,0,-2〉.〈-5,5,5〉=5+0-10=-5`

The modulus of `vecA=∥〈-1,0,-2〉∥=sqrt(1+0+4)=sqrt5`

The modulus of `vecB=∥〈-5,5,5〉∥=sqrt(25+25+25)=sqrt75`

`cos theta=(vecA.vecB)/(∥vecA∥.∥vecB∥)=-5/(sqrt5*sqrt75)=-1/sqrt15=-0.258`

`theta=105º`