# Show that the following points are collinear: A = (3, 2, -4), B = (9, 8, -10), C = (-2, -3, 1) - Mathematics and Statistics

Sum

Show that the following points are collinear:

A = (3, 2, -4), B = (9, 8, -10), C = (-2, -3, 1)

#### Solution

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

A = (3, 2, - 4), B = (9, 8, - 10) and C = (- 2, - 3, 1) respectively.

Then, bar"a" = 3hat"i" + 2hat"j" - 4hat"k",  bar"b" = 9hat"i" + 8hat"j" - 10hat"k", bar"c" = - 2hat"i" - 3hat"j" + hat"k"

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

= (9hat"i" + 8hat"j" - 10hat"k") - (3hat"i" + 2hat"j" - 4hat"k")

= 6hat"i" + 6hat"j" - 6hat"k"     ......(1)

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

= (- 2hat"i" - 3hat"j" + hat"k") - (9hat"i" + 8hat"j" - 10hat"k")

= - 11hat"i" - 11hat"j" + 11hat"k"

= - 11(hat"i" + hat"j" + 11hat"k")

=  11(hat"i" + hat"j" - hat"k")

= - 11/6 (6hat"i" + 6hat"j" - 6hat"k")

= - 11/6 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, points A, B and C are collinear.

Concept: Representation of Vector
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