Question

The vector equation of the line passing through the point \( (1,2,-4) \) and perpendicular to the two lines \[ \frac{x-8}{3} = \frac{y+19}{-16} = \frac{z-10}{7} \] and \[ \frac{x-15}{3} = \frac{y-29}{8} = \frac{z-5}{-5} \] is _____

• A. \( \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} - 3\hat{j} + 6\hat{k}) \)
• B. \( \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \)
• C. \( \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} - 6\hat{k}) \)
• D. \( \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} - 3\hat{j} - 6\hat{k}) \)

Hint:

For a line perpendicular to two lines, always take cross product of their direction vectors.

Answer 1

The Correct Option is B

Concept: Line perpendicular to two given lines → direction vector = cross product of their direction vectors.
Step 1: Direction vectors. \[ \vec{a} = (3,-16,7), \quad \vec{b} = (1,-3,6) \]
Step 2: Cross product. \[ \vec{a} \times \vec{b} = \begin{vmatrix} i & j & k
3 & -16 & 7
1 & -3 & 6 \end{vmatrix} = ( -96 +21, - (18-7), -9 +16 ) \] \[ = (-75, -11, 7) \] Simplified direction: \[ (2,3,6) \]
Step 3: Equation of line: \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \]