If direction numbers of two lines are $a, b, c$ and $b - c, c - a, a - b$ prove that they are perpendicular to each other.

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Solution

$cos θ = ∣ ∣ ∣ ∣ ∣ \frac{a (b - c) + b (c - a) + c (a - b)}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{{(b - c)}^{2} {+ (c - a)}^{2} + {(a - b)}^{2}}} ∣ ∣ ∣ ∣ ∣$
or, $cos θ = ∣ ∣ ∣ ∣ ∣ \frac{a b - a c + b c - a b + a c - b c}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{{(b - c)}^{2} {+ (c - a)}^{2} + {(a - b)}^{2}}} ∣ ∣ ∣ ∣ ∣$
or, $cos θ = 0 = cos π / 2$
$∴ θ = π / 2 \Rightarrow$ Two lines are perpendicular to each others. ...... Proved.

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