Given three non-zero, non-coplanar vectors `veca, vecb and vecc`. `vecr_1= pveca + qvecb+ vecc and vecr_2= veca + pvecb+ qvecc`. If the vectors `vecr_1 + 2vecr_2 and 2 vecr_1 + vecr_2` are collinear, then `(p, q)` is

Options

A. `(0, 0)`
B. ` (1, -1)`
C. `(-1, 1)`
D. `(1, 1)`

Answer

D

Explanation

`vecr_1 + 2 vecr_2 = (pveca + q vecb+ vecc) + 2(veca+ p vecb+ q vecc)`
`" "= (p + 2)veca + (q + 2p)vecb + (1-2q) vecc`
` 2 vecr_1 + vecr_2 = (2p +1) veca + (q+ 2p) vecb + (1 + 2q) vecc`
` 2 vecr_1 + vecr_2 = ( 2p+1) veca + ( 2q+p) vecb + ( 2+q) vecc`
` " "(p+2)/(2p+1) = (q+2p)/(2q+p) = ( 1+2q)/(2+q)`
`" " = (p+q + 2p + 2q + 3) /(p+q+2p + 2q +3) =1 `
`rArr p =1 and q=1`