Find a vector of magnitude 5 units and parallel to the resultant of the vectors `vec a=2hat i +3hat j -hat k` and `vec b=hat i-2hat j+ hat k`

Explanation

Given vectors `vec(a) = 2 hat(i) + 3 hat(j) - hat(k)`
and `vec(b) = hat(i) - 2 hat(j) + hat(k)`.
Let the resultant vector f vectors `vec(a)` and `vec(b)` is `vec(c )`.
`therefore vec(c ) = vec(a)+vec(b) = (2hat(i)+ 3hat(j) - hatk) + (hat(i) - 2hat(j) + hat(k))`
`implies vec(c)= 3hat(i) + hat(j) + 0 hat(k)`
`implies |vec(c )|=sqrt(3^(2) + 1^(2))=sqrt(9+1) = sqrt(10)`
Unit vector along `vec(c )` is `hat(c ) = (vec(c ))/(|vec(c )|) = (3hat(i)+hat(j))/(sqrt(10))`
Therefore, a vector parallel to the resultant of vectors `vec(a)` and `vec(b)` whose magnitude is 5 units is
`pm5hat(c ) = pm5 (1)/(sqrt(10))(3hat(i) + hat(j)) = pm (3sqrt(10))/(2) hat(i) pm (sqrt(10))/(2) hat(j)`