Question

If the points `P(overlinea+2overlineb+overlinec)`, `Q(2overlinea+3overlineb), R(overlineb+ t overlinec)` are collinear, where `overlinea, overlineb, overlinec` are three non-coplanar vectors, the value of t is ______ 

Options

• -2

• `-1/2`

• `1/2`

• 2

Answer 1

If the points `P(overlinea+2overlineb+overlinec)`, `Q(2overlinea+3overlineb), R(overlineb+ t overlinec)` are collinear, where `overlinea, overlineb, overlinec` are three non-coplanar vectors, the value of t is 2.

Explanation:

Since, the given points

`P(overlinea+2overlineb+overlinec)`, `Q(2overlinea+3overlineb), "and"  R(overlineb+ t overlinec)` are collinear.

∴ `overline(PQ) = lambda overline(QR)` for some scalar λ

⇒ `overlinea + overlineb - overlinec = lambda(-2overlinea - 2overlineb + t overlinec)`

⇒ `(2lambda + 1)overlinea + (1 + 2lambda)overlineb - (t lambda + 1) overlinec = overline0`

Since, `veca, vecb ,vecc` are non-coplanar vectors.

∴ `2lambda + 1 = 0, 2lambda + 1 = 0, t lambda + 1 = 0`

⇒ `lambda = -1/2` and `t = -1/lambda`

⇒ t = 2