#### 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) and (4, 3, −1).

#### Solution

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

\[\text { The direction ratios of the line joining the origin } \left( 0, 0, 0 \right) \text { to the point } \left( 2, 1, 1 \right) \text { are } \left( 2 - 0 \right), \left( 1 - 0 \right), \left( 1 - 0 \right) \text{ or } 2, 1, 1 . \]

\[ \Rightarrow a_1 = 2, b_1 = 1, c_1 = 1\]

\[\text { Similarly, the direction ratios of the line joining the points } \left( 3, 5, - 1 \right) \text { and } \left( 4, 3, - 1 \right) \text { are } \left( 4 - 3 \right), \left( 3 - 5 \right), \left[ - 1 - \left( - 1 \right) \right] \text { or } 1, - 2, 0 . \]

\[ \Rightarrow a_2 = 1, b_2 = - 2, c_2 = 0\]

\[ \therefore a_1 a_2 + b_1 b_2 + c_1 c_2 = 2 - 2 + 0 = 0\]

Therefore, the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, -1) and (4, 3, -1).