Question

Let \[ L_1 : \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{-1} \quad {and} \quad L_2 : \frac{x+1}{1} = \frac{y-2}{2} = \frac{z-2}{2} \] be two lines. Let \( L_3 \) be a line passing through the point \( (\alpha, \beta, \gamma) \) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \) where \( 5x - 11y - 8z = 1 \), then \( 5x - 11y - 8z \) equals:

Hint:

When dealing with lines and planes in 3D geometry, use vector operations like dot and cross products to find perpendicularity and compute intersection points.

Answer 1

Step 1: Extract direction ratios for \( L_1 \) as \( (1, -1, -1) \) and for \( L_2 \) as \( (1, 2, 2) \) from their parametric equations.
Step 2: Compute the cross product of the direction vectors of \( L_1 \) and \( L_2 \) to find the direction vector of \( L_3 \), which is perpendicular to both. \[ \mathbf{d_3} = \mathbf{d_1} \times \mathbf{d_2} \]
Step 3: Substitute the parametric equations of \( L_3 \) into the plane equation \( 5x - 11y - 8z = 1 \) to determine the value of \( 5x - 11y - 8z \).