Question

Show that the four points A(4, 5, 1), B(0, –1, –1), C(3, 9, 4) and D(–4, 4, 4) are coplanar.

Answer 1

Given points are A(4, 5, 1), B(0, –1, –1), C(3, 9, 4) and D(–4, 4, 4).

`∴ vec(AB) = (-4hati - 6hatj - 2hatk)`

`vec(AC) = (-hati + 4hatj + 3hatk)`

`vec(AD) = (-8hati + hatj + 3hatk)`

`∴ [(vec(AB), vec(AC), vec(AD))] = |(-4, -6, -2),(-1, 4, 3),(-8, -1, 3)|`

= – 4(12 + 3) + 6(– 3 + 24) – 2(1 + 32)

= – 60 + 126 – 66

= 0

∴ Four points A, B, C, D are coplannar.