Prove that $A (1, 2, 0), B (3, 1, 1), C (7, - 1, 3)$ are collinear.

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Solution

Consider

$\begin{matrix} - - \to A B = - - \to O B - - - \to O A = 2^i -^j +^k - - \to B C = - - \to O C - - - \to O B = 4^i - 2^j + 2^k - - \to A C = - - \to O C - - - \to O A = 6^i - 3^j + 3^k ∣ ∣ ∣ - - \to A B ∣ ∣ ∣ = \sqrt{6} ∣ ∣ ∣ - - \to B C ∣ ∣ ∣ = 2 \sqrt{6} ∣ ∣ ∣ - - \to A C ∣ ∣ ∣ = 3 \sqrt{6} ∣ ∣ ∣ - - \to A C ∣ ∣ ∣ = ∣ ∣ ∣ - - \to A B ∣ ∣ ∣ + ∣ ∣ ∣ - - \to B C ∣ ∣ ∣ \end{matrix}$

Therefore, points are collinear.

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