Question

If the lines given by \( \vec{r} = 2\hat{i} + \lambda(\hat{i} + 2\hat{j} + m\hat{k}) \) and \( \vec{r} = \hat{i} + \mu(2\hat{i} + \hat{j} + 6\hat{k}) \) are perpendicular, then the value of \( m \) is

• A. \( \dfrac{3}{2} \)
• B. \( -\dfrac{3}{2} \)
• C. \( \dfrac{2}{3} \)
• D. \( -\dfrac{2}{3} \)

Hint:

Two lines are perpendicular if the dot product of their direction vectors is zero.

Answer 1

The Correct Option is D

Step 1: Identify direction vectors.
Direction vector of first line: \[ \vec{d}_1 = (1, 2, m) \] Direction vector of second line: \[ \vec{d}_2 = (2, 1, 6) \] Step 2: Use perpendicularity condition.
For perpendicular lines, \[ \vec{d}_1 \cdot \vec{d}_2 = 0 \] Step 3: Compute dot product.
\[ (1)(2) + (2)(1) + m(6) = 0 \] \[ 2 + 2 + 6m = 0 \] Step 4: Solve for \( m \).
\[ 6m = -4 \Rightarrow m = -\frac{2}{3} \] Step 5: Conclusion.
The value of \( m \) is \( -\dfrac{2}{3} \).