# Are the four points A(1, -1, 1), B(-1, 1, 1), C(1, 1, 1) and D(2, -3, 4) coplanar? Justify your answer. - Mathematics and Statistics

Sum

Are the four points A(1, -1, 1), B(-1, 1, 1), C(1, 1, 1) and D(2, -3, 4) coplanar? Justify your answer.

#### Solution

The position vectors bar"a", bar"b", bar"c", bar"d" of the points A, B, C, D are

bar"a" = hat"i" - hat"j" + hat"k", bar"b" = -hat"i" + hat"j" + hat"k",  bar"c" = hat"i" + hat"j" + hat"k",  bar"d" = 2hat"i" - 3hat"j" + 4hat"k"

∴ bar"AB" = bar"b" - bar"a"

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

= - 2hat"i" + 2hat"j"

bar"AC" = bar"c" - bar"a"

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

and bar"AD" = bar"d" - bar"a" = (2hat"i" - 3hat"j" + 4hat"k") - (hat"i" - hat"j" + hat"k")

= hat"i" - 2hat"j" + 3hat"k"

If A, B, C, D are coplanar, then there exist scalars x, y such that

bar"AB" = "x".bar"AC" + "y".bar"AD"

∴ - 2hat"i" + 2hat"j" = "x"(2hat"j") + "y"(hat"i" - 2hat"j" + 3hat"k")

∴ - 2hat"i" + 2hat"j" = "y"hat"i" + (2"x" - 2"y")hat"j" + "3y"hat"k"

By equality of vectors,

y = - 2     ....(1)

2x - 2y = 2     .....(2)

3y = 0     ....(3)

From (1), y = - 2

From (3), y = 0

This is not possible.

Hence, the points A, B, C, D are not coplanar.

Concept: Representation of Vector
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