#### Question

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), (4, 3, – 1).

#### Solution

Let OA be the line joining the origin, O (0, 0, 0), and the point, A (2, 1, 1).

Also, let BC be the line joining the points, B (3, 5, −1) and C (4, 3, −1).

The direction ratios of OA are 2, 1, and 1 and of BC are (4 − 3) = 1, (3 − 5) = −2, and (−1 + 1) = 0

OA is perpendicular to BC, if *a*_{1}*a*_{2} + *b*_{1}*b*_{2} + *c*_{1}*c*_{2} = 0

∴ *a*_{1}*a*_{2} + *b*_{1}*b*_{2} + *c*_{1}*c*_{2} = 2 × 1 + 1 (−2) + 1 ×0 = 2 − 2 = 0

Thus, OA is perpendicular to BC.

Is there an error in this question or solution?

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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), (4, 3, – 1). Concept: Equation of a Line in Space.

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