Show that
$A (1, - 2, - 8), B (5, 0, - 2)$ and $C (11, 3, 7)$ are collinear.

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Solution

The given points are A( $1, - 2, - 8$ ), B( $5, 0, - 2$ ), C( $11, 3, 7$ )
$\begin{matrix} ∴ - - \to A B = (5 - 1) ˆ i + (0 + 2) ˆ j + (- 2 + 8) ˆ k = 4 ˆ i + 2 ˆ j + 6 ˆ k - - \to B C = (11 - 5) ˆ i + (3 - 0) ˆ j + (7 + 2) ˆ k = 6 ˆ i + 3 ˆ j + 9 ˆ k - - \to A C = (11 - 1) ˆ i + (3 + 2) ˆ j + (7 + 8) ˆ k = 10 ˆ i + 5 ˆ j + 15 ˆ k ∣ ∣ ∣ - - \to A B ∣ ∣ ∣ = \sqrt{4^{2} + 2^{2} + 6^{2}} = \sqrt{16 + 4 + 36} = \sqrt{56} = 2 \sqrt{14} ∣ ∣ ∣ - - \to B C ∣ ∣ ∣ = \sqrt{6^{2} + 3^{2} + 9^{2}} = \sqrt{36 + 9 + 81} = \sqrt{126} = 3 \sqrt{14} ∣ ∣ ∣ - - \to A C ∣ ∣ ∣ = \sqrt{10^{2} + 5^{2} + 15^{2}} = \sqrt{100 + 25 + 225} = \sqrt{350} = 5 \sqrt{14} ∴ ∣ ∣ ∣ - - \to A C ∣ ∣ ∣ = ∣ ∣ ∣ - - \to A B ∣ ∣ ∣ + ∣ ∣ ∣ - - \to B C ∣ ∣ ∣ \end{matrix}$
Thus, the given points A,B, and C are collinear.

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