Question

The direction ratios of two straight lines are \( l, m, n \) and \( l_1, m_1, n_1 \). The lines will be perpendicular to each other if:

• A. \( \frac{l_1}{l} = \frac{m_1}{m} = \frac{n_1}{n} \)
• B. \( \frac{l_1}{l} + \frac{m_1}{m} + \frac{n_1}{n} = 0 \)
• C. \( l^2 + m^2 + n^2 = l_1^2 + m_1^2 + n_1^2 \)
• D. \( l l_1 + m m_1 + n n_1 = 0 \)

Hint:

To check if two lines are perpendicular, compute the dot product of their direction ratios. If the dot product is zero, the lines are perpendicular.

Answer 1

The Correct Option is D

For two straight lines to be perpendicular, the dot product of their direction ratios must be zero. Let the direction ratios of the first line be \( (l, m, n) \) and the direction ratios of the second line be \( (l_1, m_1, n_1) \). The dot product of the two direction ratios is given by: \[ l l_1 + m m_1 + n n_1 = 0 \] This condition must hold for the lines to be perpendicular. Thus, the correct condition for the lines to be perpendicular is: \[ l l_1 + m m_1 + n n_1 = 0 \] Hence, the correct answer is: \[ \boxed{l l_1 + m m_1 + n n_1 = 0} \]