Show that following points are collinear P(4, 5, 2), Q(3, 2, 4), R(5, 8, 0)

#### Solution

Let `bar("p"), bar("q"), bar("r")` be the position vectors of points P, Q, R respectively.

∴ `bar("p") = 4hat"i" + 5hat"j" + 2hat"k"`,

`bar("q") = 3hat"i" + 2hat"j" + 4hat"k"`,

`bar("r") = 5hat"i" + 8hat"j"`

∴ `bar("PQ") = bar("q") - bar("p")`

= `(3hat"i" + 2hat"j" + 4hat"k") - (4hat"i" + 5hat"j" + 2hat"k")`

∴ `bar("PQ") = hat"i" - 3hat"j" + 2hat"k"`

`bar("QR") = bar("r") - bar("q")`

= `5hat"i" + 8hat"j" - (3hat"i" + 2hat"j" + 4hat"k")`

= `2hat"i" + 6hat"j" - 4hat"k"`

∴ `bar("QR") = (-2) (-hat"i" - 3hat"j" + 2hat"k")` .......(ii)

∴ `bar("QR") = (-2) bar("PQ")` .......[From (i) and (ii)]

∴ `bar("QR")` is a scalar multiple of `bar("PQ")`

∴ `bar("QR")` and `bar("PQ")` are parallel to each other with point Q in common.

∴ `bar("PQ")` and `bar("QR")` lie on the same line.

∴ Points P, Q and R are collinear.