Question

The value of \( \lambda \) for which the angle between lines \( \vec{r} = \hat{i} + \hat{j} + \hat{k} + p(2\hat{i} + \hat{j} + 2\hat{k}) \) and \( \vec{r} = (1+q)\hat{i} + (1+q\lambda)\hat{j} + (1+q)\hat{k} \) is \( \frac{\pi}{2} \)

• A. -4
• B. 2
• C. -2
• D. 4

Hint:

For angle between lines, use direction vectors and apply dot product condition.

Answer 1

The Correct Option is A

Step 1: Identify direction vectors.
From first line:
\[ \vec{d_1} = (2,1,2). \]
From second line:
\[ \vec{d_2} = (1,\lambda,1). \]

Step 2: Use condition for perpendicular lines.
Two lines are perpendicular if:
\[ \vec{d_1} \cdot \vec{d_2} = 0. \]

Step 3: Compute dot product.
\[ (2,1,2)\cdot(1,\lambda,1). \]
\[ = 2(1) + 1(\lambda) + 2(1). \]
\[ = 2 + \lambda + 2. \]
\[ = \lambda + 4. \]

Step 4: Apply perpendicular condition.
\[ \lambda + 4 = 0. \]

Step 5: Solve for \( \lambda \).
\[ \lambda = -4. \]

Step 6: Verify concept.
Dot product zero ensures angle between lines is \(90^\circ\).

Step 7: Final conclusion.
Thus, the required value is:
\[ \boxed{-4}. \]