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

Concept:
Two planes are parallel if their normal vectors are parallel. So if \[ \vec{n}_1=(2,-\lambda,1) \quad \text{and} \quad \vec{n}_2=(4,-1,\mu), \] then for parallel planes: \[ (2,-\lambda,1)=k(4,-1,\mu) \] for some scalar \(k\). ip

Step 1: Equate the corresponding components.
From the first component: \[ 2 = 4k \] \[ k = \frac{1}{2} \] ip

Step 2: Find \(\lambda\).
From the second component: \[ -\lambda = -1 \cdot \frac{1}{2} \] \[ -\lambda = -\frac{1}{2} \] \[ \lambda = \frac{1}{2} \] ip

Step 3: Find \(\mu\).
From the third component: \[ 1 = \mu \cdot \frac{1}{2} \] \[ \mu = 2 \] ip

Step 4: Calculate \(\lambda+\mu\).
\[ \lambda+\mu = \frac{1}{2}+2 = \frac{5}{2} \] ip Hence, the correct answer is:
\[ \boxed{(C)\ \frac{5}{2}} \]