Question

Show that the following points are collinear:

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

Answer 1

Let `bar"a" , bar"b" , bar"c"` be position vectors of the points.

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

Then `bar"a" = 4hat"i" + 5hat"j" + 2hat"k" ,  bar"b" = 3hat"i" + 2hat"j" + 4hat"k" ,  bar"c" = 5hat"i" + 8hat"j" + 0hat"k"`

`bar"AB" = bar"b" - bar"a"`

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

`= - hat"i" - 3hat"j" + 2hat"k"` i.e.

`= - (hat"i" + 3hat"j" - 2hat"k")`      .....(1)

and `bar"BC" = bar"c" - bar"b"`

`= (5hat"i" + 8hat"j" + 0hat"k") - (3hat"i" + 2hat"j" + 4hat"k")`

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

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

`= 2 . bar"AB"`           ....[By(1)]

∴ `bar"BC"` is a non-zero scalar multiple of `bar"AB"`

∴ they are parallel to each other.

But they have point B in common.

∴ `bar"BC"`  and  `bar"AB"` are collinear vectors.

Hence, the points A, B, and C are collinear.