Question

If the points A(1, 2, 3), B(–1, 1, 2), C(2, 3, 4) and D(–1, x, 0) are coplanar find the value of x.

Answer 1

A = (1, 2, 3),

B = (–1, 1, 2),

C = (2, 3, 4),

D = (–1, x, 0).

∴ `bar(AB) = barb - bara = -2hati - hatj - hatk`,

`bar(AC) = barc - bara = hati + hatj + hatk`

`bar(AD) = bard - bara = -2hati + (x - 2)hatj - 3hatk`

Since the points A, B, C, D are coplanar, the vectors `bar(AB), bar(AC), bar(AD)` are coplanar.

∴ `[(bar(AB), bar(AC), bar(AD))]` = 0

∴ `|(-2, -1, -1),(1, 1, 1),(-2, (x - 2), -3)|` = 0

∴ – 2(–3 – x + 2) + 1(–3 + 2) – 1(x – 2 + 2) = 0

∴ – 2(– x – 1) – 1 – (x) = 0

∴ 2x + 2 – 1 – x = 0

∴ x = – 1