#### 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?

Solution 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.