Question

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

Answer 1

Please see the explanation.

Explanation:

The scalar product as two definitions:

`"[1] "barA*barB = (a_x)(b_x) + (a_y)(b_y) + (a_z)(b_z)`
`"[2] "barA*barB = ||barA||||barB||cos(theta)`

where `theta` is the angle between the vectors.

Use [1] to compute the scalar product:

`barA*barB = (3)(4) + (4)(-5) + (-4)(5)`

`barA*barB = -28`

Compute the magnitude of `barA and barB`

`||barA|| = sqrt(3^2 + 4^2 + (-4)^2)`

`||barA|| = sqrt(41)`

`||barB|| = sqrt(4^2 + (-5)^2 + 5^2)`

`||barB|| = sqrt(66)`

Use [2] to find the value of `theta`

`"[2] "barA*barB = ||barA||||barB||cos(theta)`

`-28= sqrt(41)sqrt(66)cos(theta)`

`-28/(sqrt(41)sqrt(66)) = cos(theta)`

`theta = cos^-1(-28/(sqrt(41)sqrt(66)))`

`theta ~~ 122.565^@`