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).

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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