If a unit vector `vec(a)` makes angles `.^(pi)//_(4)` with `hat(i), .^(pi)//_(3)` with `hat(j)` and an acute angle `theta` with `hat(k)` then find the value of `theta`. Also, find the scalar and vector components of a along the axes.

Explanation

Let `vec(a)=(a_(1) hat(i)+a_(2) hat(j)+a_(3) hat(k))`. Then, `vec(a)` being a unit vector, we have `(a_(1)^(2)+a_(2)^(2)+a_(3)^(2))=1`.
Now, `vec(a). Hat(i) = a_(1) implies |vec(a)||hat(i)|"cos" pi/4=a_(1) implies a_(1) = 1/sqrt(2) [ :' |vec(a)|=1, |hat(i)|=1]`
`vec(a).hat(j)=a_(1) implies |vec(a)||hat(j)| "cos" pi/3=a_(2) implies a_(2)=1/2 [ :' |vec(a)|=1, |hat(j)|=1]`
`vec(a).hat(k)=a_(3) implies |vec(a)||hat(k)| cos theta=a_(3) implies a_(3) = cos theta [ :' |vec(a)|=1, |hat(k)|=1]`
Now, `|vec(a)|=1implies |vec(a)|^(2) =1`
`implies a_(1)^(2)+a_(2)^(2)+a_(3)^(2)=1`
`implies 1/2+1/4 + cos^(2) theta =1 [ :' a_(1)=1/sqrt(2), a_(2) =1/2, a_(3) = cos theta]`
`implies cos^(2) theta=1/4 implies cos theta =1/2 implies theta = pi/3`.
Hence, the scalar components of `vec(a)` are `1/sqrt(2), 1/2` and `1/2`.
and, the vector components of `vec(a)` are `1/sqrt(2) hat(i), 1/2 hat(j)` and `1/2 hat(k)`.