Q. 24.2( 54 Votes )

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).

Answer :

We know that

Two lines with direction ratios a1, b1, c1 and a2, b2, c2 are perpendicular if the angle between them is θ = 90°, i.e. a1a2 + b1b2 + c1c2 = 0


Also, we know that the direction ratios of the line segment joining (x1, y1, z1) and (x2, y2, z2) is taken as x2 – x1, y2 – y1, z2 – z1 (or x1 – x2, y1 – y2, z1 – z2).


The direction ratios of the line through the points (1, –1, 2) and (3, 4, –2) is:


a1 = 3 – 1 = 2, b1 = 4 – (-1) = 4 + 1 = 5, c1 = -2 –2 = -4


and the direction ratios of the line through the points (0, 3, 2) and (3, 5, 6) is:
a2 = 3 – 0 = 3, b2 = 5 – 3 = 2, c2 = 6 – 2 = 4


Now, consider


a1a2 + b1b2 + c1c2 = 2 × 3 + 5 × 2 + (-4) × 4 = 6 + 10 + (-16) = 16 + (-16) = 0


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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