Show that the line through the points (1, −1, 2) (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

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#### Solution

Let AB be the line joining the points, (1, −1, 2) and (3, 4, − 2), and CD be the line joining the points, (0, 3, 2) and (3, 5, 6).

The direction ratios, *a*_{1}, *b*_{1}, *c*_{1}, of AB are (3 − 1), (4 − (−1)), and (−2 − 2) i.e., 2, 5, and −4.

The direction ratios, *a*_{2}, *b*_{2}, *c*_{2}, of CD are (3 − 0), (5 − 3), and (6 −2) i.e., 3, 2, and 4.

AB and CD will be perpendicular to each other, 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 × 3 + 5 × 2 + (− 4) × 4

= 6 + 10 − 16

= 0

Therefore, AB and CD are perpendicular to each other.

Concept: Equation of a Line in Space

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