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Question

Find x such that the four points A(4, 1, 2), B(5, x, 6), C(5, 1, 1) and D(7, 4, 0) are coplanar.

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Solution

AB=^i+(x1)^j+4^k

BC=0^i+(1x)^j7^k

CD=2^i+3^j+^k

Since the points A, B, C, D are coplanar

[AB BC CD]=0

∣ ∣1x1401x7231∣ ∣=0

1(1x+21)+2(7x+74+4x)=0

(22x)+2(33x)=0

22x+66x=0

7x=28

x=4

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