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 a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are perpendicular if the angle between them is θ = 90°, i.e. a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

Also, we know that the direction ratios of the line segment joining (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is taken as x_{2} – x_{1}, y_{2} – y_{1}, z_{2} – z_{1} (or x_{1} – x_{2}, y_{1} – y_{2}, z_{1} – z_{2}).

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

a_{1} = 3 – 1 = 2, b_{1} = 4 – (-1) = 4 + 1 = 5, c_{1} = -2 –2 = -4

and the direction ratios of the line through the points (0, 3, 2) and (3, 5, 6) is:

a_{2} = 3 – 0 = 3, b_{2} = 5 – 3 = 2, c_{2} = 6 – 2 = 4

Now, consider

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 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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