Question

Show that the following points are collinear:

P = (4, 5, 2), Q = (3, 2, 4), R = (5, 8, 0).

Answer 1

Let `bar"p", bar"q", bar"r"` be position vectors of the points.

P = (4, 5, 2), Q = (3, 2, 4), R = (5, 8, 0) respectively.

Then `bar"p" = 4hati + 5hatj + 2hatk,  bar"q" = 3hati + 2hatj + 4hatk,  bar"r" = 5hati + 8hatj + 0hatk`

`bar("PQ") = bar"q" - bar"p"`

= `(3hati + 2hatj + 4hatk) - (4hati + 5hatj + 2hatk)`

= `-hati - 3hatj + 2hatk` 

= `-(hati + 3hatj - 2hatk)`      .....(1)

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

= `(5hati + 8hatj + 0hatk) - (3hati + 2hatj + 4hatk)`

= `2hati + 6hatj - 4hatk`

= `2(hati + 3hatj - 2hatk)`

= `2.bar("PQ")`           ....[By(1)]

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

∴ They are parallel to each other.

But they have point Q in common.

∴ `bar("PQ")`  and  `bar("QR")` are collinear vectors.

Hence, the points P, Q, and R are collinear.