Question

Find the angle between the vectors `hati - 2hatj + 3hatk` and `3hati - 2hatj + hatk`.

Answer 1

Let, `vec"a" = (hati - 2hatj + 3hatk)`

and `vec"b" = (3hati - 2hatj + hatk)`

Then `|veca| = sqrt(1^2 + (-2)^2 + 3^2)`

`= sqrt(1 + 4 + 9)`

`= sqrt14`

`|vecb| = sqrt(3^2 + (-2)^2 + 1^2)`

`= sqrt (9 + 4 + 1)`

`= sqrt14`

and `veca . vecb = (hati - 2hatj + 3hatk) * (3hati - 2hatj + hatk)`

= (1) (3) + (-2) (-2) + (3) (1)

= 3 + 4 + 3

= 10

If θ be the angle between `veca  "and"  vecb`, then

`cos theta = (veca . vec b)/(|veca| |vec b|)`

`= 10/(sqrt14 sqrt14)`

`= 10/14`

`= 5/7`

⇒ `theta = cos ^-1  5/7`