With respect to a rectangular Cartesian co-ordinate system, three vectors are expressed as `vec a = 4hati -hatj, vec b =-3hati +2hatj, vec c=-hatk` where `hati, hatj, hatk` are unit vectors, along the x, y and z-axes respectively. The unit vector `hatr` along the direction of the sum of these three vectors is given by

Options

A. `hatr = (1)/(sqrt(3))(hati+hatj-hatk)`
B. `hatr =(1)/(sqrt(2)) (hati+hatj-hatk)`
C. `hatr =(1)/(3) (hati+hatj-hatk)`
D. `hatr=(1)/(sqrt(3))(hati+hatj+hatk)`

Answer

A

Explanation

The sum of the three vectors
`= vec a + vec b + vec c`
`=(4hati - hatj) +(-3hatj + 2hatj) + (-veck)`
`=hati + hatj -hatk`
`therefore | hati + hatj - hatk|= sqrt(1^(2) + 1^(2) +1^(2)) = sqrt(3)`
`therefore` The unit vecotr `hatr` along the direction of the sum of these three vectors is given by
`hatr =(hati + hatj - hatk)/(|hati + hatj -hatk|) =(hati + hatj -hatk)/(sqrt(3))`