# Show that the following points are collinear: P = (4, 5, 2), Q = (3, 2, 4), R = (5, 8, 0). - Mathematics and Statistics

Sum

Show that the following points are collinear:

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

#### Solution

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.

Concept: Representation of Vector
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