The points with position vectors `alpha hati+hatj+hatk, hati-hatj-hatk, hati+2hatj-hatk, hati+hatj+betahatk` are coplanar if

Options

A. `(1-alpha)(1+beta)=0`
B. `(1-alpha)(1-beta)=0`
C. `(1+alpha)(1+beta)=0`
D. `(1+alpha)(1-beta)=0`

Answer

A

Explanation

Let `P,Q,R` and `S` be the given points with position vectors `alpha hati+hatj+hatk, hati-hatj-hatk, hati+2hatj-hatk` and `hati+hatj+betahatk` respectively. Then,
`vec(QP)=(alpha-1)hati+2hatj+2hatk, vec(QR)=0hati+3hatj+0hatk`
and `vec(QS)=0hati+2hatj+(beta+1)hatk` are coplanar.
`:.[vec(QP),vec(QR),vec(QS)]=0`
`implies|(alpha-1,2,2),(0,3,0),(0,2,beta+1)|=0`
`implies (alpha-1)(beta+1)=0implies(1-alpha)(1+beta)=0`