Discussion
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
Answer: E
\(\lambda = -4\) and \(12\).
\[ \begin{aligned} \vec{a} + \vec{b} + \vec{c} &= 0\hat{i} - 2\hat{j} + (12 - \lambda)\hat{k} \\ \vec{a} - \vec{b} + \vec{c} &= 10\hat{i} + 0\hat{j} + (-4 - \lambda)\hat{k} \end{aligned} \]
Perpendicular vectors have zero dot product:
\[ \begin{aligned} 0 + 0 + (12 - \lambda)(-4 - \lambda) &= 0 \\ \lambda &= 12 \text{ or } \lambda = -4 \end{aligned} \]
The option 4 and -12 has both signs reversed, and the other pairs do not make either factor zero.
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