Question

Let $\vec{a} = 2\hat{i} - 2\hat{j} + 4\hat{k}$, $\vec{b} = -5\hat{i} - \hat{j} + 8\hat{k}$ and $\vec{c} = 3\hat{i} + \hat{j} - \lambda \hat{k}$. If $\vec{a} + \vec{b} + \vec{c}$ and $\vec{a} - \vec{b} + \vec{c}$ are perpendicular, then the values of $\lambda$ are:

• A. $4$ and $-12$
• B. $-2$ and $12$
• C. $-6$ and $14$
• D. $-3$ and $12$
• E. $-4$ and $12$

Hint:

Perpendicular vectors $\Rightarrow$ dot product equals zero.

Answer 1

Answer: (E)

Concept:
• Perpendicular vectors $\Rightarrow$ dot product $=0$

Step 1: Compute vectors
\[ \vec{a} + \vec{b} + \vec{c} = (2-5+3,\,-2-1+1,\,4+8-\lambda) = (0,-2,12-\lambda) \] \[ \vec{a} - \vec{b} + \vec{c} = (2+5+3,\,-2+1+1,\,4-8-\lambda) = (10,0,-4-\lambda) \]

Step 2: Apply dot product
\[ (0,-2,12-\lambda) \cdot (10,0,-4-\lambda) = 0 \] \[ 0 + 0 + (12-\lambda)(-4-\lambda) = 0 \]

Step 3: Solve equation
\[ (12-\lambda)(-4-\lambda) = 0 \] \[ \lambda = 12 \quad \text{or} \quad \lambda = -4 \] Final Conclusion:
Option (E)