Let `veca=hati + 2hatj +hatk, vecb=hati - hatj +hatk andvecc= hathatj-hatk` A vector in the plane of `veca and vecb` whose projections on `vecc is 1//sqrt3` is

Options

A. `hati+hatj-2hatk`
B. `3hati+hatj-3hatk`
C. `4hati-hatj+4hatk`
D. `4hati+hatj-4hatk`

Answer

C

Explanation

Any vector r in the plane containing a and b is `a + lambda b`
`therefore r = ( hati + 2 hatj + hatk ) + lambda(hati-hatj+hatk)`
`=(1+lambda)hati + (2 -lambda)hatj + (1+lambda)hatk`
Projection of r on `c=(r*c)/(|c|)`
`rArr ([(1+lambda)hati+(2-lambda)hatj+(1+lambda)hatk]*(hati+hatj-hatk))/(sqrt((1)^(2)+(1)^(2)+(-1)^(2)))=|(1)/(sqrt(3))|`
`rArr (1+lambda + 2 -lambda - 1-lambda)/(sqrt(3))=|(1)/(sqrt(3))|`
`rArr 2-lambda = pm 1 rArr lambda 1,3 `
`therefore r = 2 hati +hatj + 2 hatk , 4 hati -hatj + 4 hatk`