Find the angle between the vectors `vec(a)` and `vec(b)`, when
(i) `vec(a)=hat(i)-2hat(j)+3 hat(k)` and `vec(b)=3hat(i)-2hat(j)+hat(k)`
(ii) `vec(a)=3 hat(i)+hat(j)+2hat(k)` and `vec(b)=2hat(i)-2hat(j)+4 hat(k)`
(iii) `vec(a)=hat(i)-hat(j)` and `vec(b)=hat(j)+hat(k)`.

Answer

(i) `cos^(-1) (5/7)` (ii) `cos^(-1) (sqrt(3/7))` (iii) `120^(@)`

Explanation

(iii) Let the required angle be `theta`. Then,
`vec(a).vec(b)=(1xx0)+(-1)xx1+(0xx1)=-1`
`implies |vec(a)||vec(b)| cos theta=-1`
`implies (sqrt(2))(sqrt(2)) cos theta=-1 implies cos theta= - 1/2 implies theta = 120^(@)`.