With respect to a rectangular Cartesian coordinate system, three vectors are expressed as
`vec(a)= 4hat(i)-hat(j), vec(b)= -3hat(i)+2hat(j) and vec(c )= -hat(k)`
Where, `hat(i), hat(j), hat(k)` are unit Vector, along the X, Y and Z-axis respectively. The unit vectors `hat(r )` along the direction of sum of these vector is

Options

A. `hat(r) = (1)/(sqrt3) (hat(i) + hat(j)- hat(k))`
B. `hat(r) = (1)/(sqrt2) (hat(i) + hat(j) - hat(k))`
C. `hat(r) = (1)/(3) (hat(i)- hat(j)- hat(k))`
D. `hat(r)= (1)/(sqrt2) (hat(i) + hat(j) + hat(k))`

Answer

A

Explanation

`vec(r)= vec(a) + vec(b) + vec(c) = 4 hat(i)- hat(j)- 3hat(i) + 2hat(j)- hat(k)= hat(i) + hat(j) - hat(k)`
Unit vector `hat(r)= (vec(r))/(|r|)= (hat(i) + hat(j) - hat(k))/(sqrt(1^(2) + 1^(2) + (-1)^(2)))= (hat(i) + hat(j) - hat(k))/(sqrt3)`