Question

If the line \[ \frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1} \] is parallel to the plane \[ a_2 x + b_2 y + c_2 z + d = 0 \] then:

• A. \( \frac{a_2}{a_1} = \frac{b_2}{b_1} = \frac{c_2}{c_1} \)
• B. \( a_1 x + b_1 y + c_1 z = 0 \)
• C. \( a_1 a_2 + b_1 b_2 + c_1 c_2 = 0 \)
• D. none of these

Hint:

For a line to be parallel to a plane, the dot product of the line’s direction ratios and the plane’s normal vector must be zero: \[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 0 \]

Answer 1

The Correct Option is C

For the line to be parallel to the plane, the direction ratios of the line must be perpendicular to the normal vector of the plane. The direction ratios of the line are \( (a_1, b_1, c_1) \), and the normal vector to the plane is \( (a_2, b_2, c_2) \). The condition for perpendicularity is given by the dot product of the direction ratios of the line and the normal vector of the plane being zero: \[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 0 \] Thus, the correct option is: \[ \boxed{a_1 a_2 + b_1 b_2 + c_1 c_2 = 0} \]