Find a vector `vec a` of magnitude `5sqrt(2)` making an angle `pi/4` with x-axis , `pi/2` with y-axis and an acute angle `theta` with z-axis

Answer

`5(hat(i)+hat(k))`

Explanation

Let `vec(a)=a_(1) hat(i) +a_(2) hat(j) + a_(3) hat(k)`. Then, `|vec(a)|^(2)=50implies (a_(1)^(2)+a_(2)^(2)+a_(3)^(2))=50`.
`vec(a).hat(i)=|vec(a)||hat(i)| cos alpha implies a_(1) = sqrt(50) xx1xx "cos" pi/4=5 sqrt(2) xx1/sqrt(2)=5`.
`vec(a).hat(j)=|vec(a)||hat(j)| cos beta implies a_(2) = sqrt(50)xx1xx"cos" pi/2=0`
`vec(a).hat(k)=|vec(a)||hat(k)| cos gamma implies a_(3) = sqrt(50)xx1xxcos theta`
`(a_(1)^(2)+a_(2)^(2)+a_(3)^(2))=50 implies 25+0+50 cos^(2) theta=50`
`implies cos^(2) theta =1/2 implies cos theta = 1/sqrt(2) = gt theta= pi/2`.
`:. a_(3)=sqrt(50)xx "cos" pi/4=(5sqrt(2)xx1/sqrt(2))=5`.
`:. vec(a)=5(hat(i)+hat(k))`.