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 $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$
$\therefore a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=2 \times 1+1(-2)+1 \times 0=2-2=0$
Thus, OA is perpendicular to BC.
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