Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).

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Solution

The direction ratios of the line joining the origin to the point (2, 1, 1) are 2, 1, 1.

Let $\vec{b_{1}} = 2 \hat{i} + \hat{j} + \hat{k}$

The direction ratios of the line joining the points (3, 5, $-$ 1) and (4, 3, $-$ 1) are 1, $-$ 2, 0.

Let $\vec{b_{2}} = \hat{i} - 2 \hat{j} + 0 \hat{k}$

Now,

$\vec{b_{1}} . \vec{b_{2}} = (2 \hat{i} + \hat{j} + \hat{k}) . (\hat{i} - 2 \hat{j} + 0 \hat{k}) = 2 - 2 + 0 = 0 ∴ \vec{b_{1}} ⊥ \vec{b_{2}}$

Hence, the two lines joining the given points are perpendicular to each other.

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