#### Question

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

#### Solution

\[\text { We know that two lines with direction ratios } a_1 , b_1 , c_1 \text { and } a_2 , b_2 , c_2 \text { are perpendicular if } a_1 a_2 + b_1 b_2 + c_1 c_2 = 0 . \]

\[\text { The direction ratios of the line passing through the points }\left( 1, - 1, 2 \right) \text{ and } \left( 3, 4, - 2 \right) \text{ are } \left( 3 - 1 \right), \left[ 4 - \left( - 1 \right) \right], \left( - 2 - 2 \right), \text { i . e } . 2, 5, - 4 . \]

\[ \Rightarrow a_1 = 2, b_1 = 5, c_1 = - 4\]

\[\text { Similarly, the direction ratios of the line passing through the points } \left( 0, 3, 2 \right) \text { and } \left( 3, 5, 6 \right) \text { are }\left( 3 - 0 \right), \left( 5 - 3 \right), \left( 6 - 2 \right), \text{ i . e} . 3, 2, 4 . \]

\[ \Rightarrow a_2 = 3, b_2 = 2, c_2 = 4\]

\[ \therefore a_1 a_2 + b_1 b_2 + c_1 c_2 = 2 \times 3 + 5 \times 2 + \left( - 4 \right) \times 4 = 6 + 10 - 16 = 0\]

Thus, the line through the points (1, -1, 2) and (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6)