`veca=2hati-hatj+hatk,vecb=hatj+2hatj-hatk,vecc=hati+hatj -2 hatk` . A vector coplanar with `vecb and vecc`. Whose projection on `veca` is magnitude `sqrt(2/3)` is

Options

A. `2hati+3hatj-3hatk`
B. `-2hati-hatj+5hatk`
C. `2hati+3hatj+3hatk`
D. `2hati+hatj+5hatk`

Answer

b

Explanation

Let the required vector be `vecr`. Then
`vecr = x_(1) vecb + x_(2) vecc and vecr .veca = sqrt(2/3) " " (|veca|)=2`
Now ` vecr.veca = x_(1) veca. Vecb + x_(2) veca.vecc`
` Rightarrow 2 = x_(1) ( 2-2-1) + x_(2) ( 2-1-2)`
` Rightarrow x_(1) + x_(2) = -2`
` Rightarrow vecr = x_(1) ( hati + 2hatj -hatk) + x_(2) (hati +hatj - 2hatk)`
`= hati (x_(1) + x_(2) +hatj ( 2x_(1) + x_(2) ) -hatk ( 2x_(2) +x_(1))`
` 2hati + hatj (x_(1) -2) -hatk(-4-x_(1))`,