Question

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.

Answer 1

The position vectors `bara, barb, barc, bard` of the points A, B, C, D are;

`bara = hati - hatj + hatk, barb = -hati + hatj + hatk, barc = hati + hatj + hatk,  bard = 2hati - 3hatj + 4hatk`

∴ `bar(AB) = barb - bara`

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

`= - 2hati + 2hatj`

`bar(AC) = barc - bara`

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

`= 2hatj`

and `bar(AD) = bard - bara`

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

`= hati - 2hatj + 3hatk`

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

`bar(AB) = x.bar(AC) + y.bar(AD)`

∴ `- 2hati + 2hatj = x(2hatj) + y(hati - 2hatj + 3hatk)`

∴ `- 2hati + 2hatj = yhati + (2x - 2y)hatj + 3yhatk`

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.