Question

If the planes \(\bar{r} \cdot (2\hat{i} - \lambda\hat{j} + \hat{k}) = 3\) and \(\bar{r} \cdot (4\hat{i} - \hat{j} + \mu\hat{k}) = 5\) are parallel, then \(\lambda + \mu =\)

• A. \(\frac{1}{2}\)
• B. \(2\)
• C. \(\frac{5}{2}\)
• D. \(\frac{7}{2}\)

Hint:

For parallel planes, compare their normal vectors, not the constants on the right-hand side.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Two planes are parallel if their normal vectors are proportional.
Step 2: Key Formula or Approach:
Normal vectors \(\bar{n}_1 = (2, -\lambda, 1)\) and \(\bar{n}_2 = (4, -1, \mu)\).
Condition: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).
Step 3: Detailed Explanation:
Equating ratios:
\[ \frac{2}{4} = \frac{-\lambda}{-1} = \frac{1}{\mu} \] \[ \frac{1}{2} = \lambda \implies \lambda = \frac{1}{2} \] \[ \frac{1}{2} = \frac{1}{\mu} \implies \mu = 2 \] Now, \(\lambda + \mu = \frac{1}{2} + 2 = \frac{5}{2}\).
Step 4: Final Answer:
The sum is \(\frac{5}{2}\).