Discussion
Let \(\vec{a} + \vec{b} = \lambda\hat{i} + 16\hat{j} - 18\hat{k}\) and \(\vec{a} - \vec{b} = 2\hat{i} + 8\hat{j} + \lambda\hat{k}\). If \(\vec{a} + \vec{b}\) is perpendicular to \(\vec{a} - \vec{b}\),then \(|\vec{a}| =\)
Answer: E
\(|\vec{a}| = \) \(\sqrt{194}\).
Perpendicular vectors have zero dot product:
\[ \begin{aligned} (\vec{a} + \vec{b})\cdot(\vec{a} - \vec{b}) &= 2\lambda + 128 - 18\lambda = 0 \\ \lambda &= 8 \end{aligned} \]
Adding the two given vectors and halving gives \(\vec{a}\):
\[ \begin{aligned} \vec{a} &= \tfrac{1}{2}(10\hat{i} + 24\hat{j} - 10\hat{k}) = 5\hat{i} + 12\hat{j} - 5\hat{k} \\ |\vec{a}| &= \sqrt{25 + 144 + 25} = \sqrt{194} \end{aligned} \]
The other surds come from a wrong \(\lambda\) or from taking \(|\vec{a} + \vec{b}|\) instead of \(|\vec{a}|\).
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