If a unit vector `hat(a)`, makes angles `(pi)/(3)` with `hat(i),(pi)/(4)` with `hat(j)` and an acute angle `theta` with `hat(k)`, then find `theta` and hence the components of a.

Explanation

Let unit vector makes angle `alpha, beta and gamma ` with `hat(i),hat(j), and hat(k)` respectively, then `alpha=(pi)/(3),beta=(pi)/(4)and gamma=theta`.
`therefore cos^2(pi)/(3)+cos^2(pi)/(4)+cos^2 theta=1rArr ((1)/(2))^2+((1)/(sqrt(2)))^2+cos^2 theta=1`
`rArr (1)/(4)+(1)/(2)+cos^2 theta=1rArr cos^2 theta=1-(3)/(4) rArr cos^2 theta=(4-3)/(4)=(1)/(4)`
`rArr cos theta = +-(1)/(sqrt(4))rArr cos theta = +-(1)/(2)`
`cos theta =(1)/(2)`
`rArr theta = cos^(-1) ((1)/(2))= cos^(-1)(cos.(pi)/(3))`
`rArr theta=(pi)/(3)` and components of a are `cos. (pi)/(3), cos. (pi)/(4), cos.(pi)/(3)`.
`rArr (1)/(2),(1)/(sqrt(2)),(1)/(2)`.