Discussion
Let \(P(2, 1, -3)\) be a given point. If the point \(Q\) is such that \(\overrightarrow{PQ} = 3\hat{i} - \hat{j} + 5\hat{k}\), then the distance of \(Q\) from the origin \(O\) is
Answer: A
The distance is \(\sqrt{29}\).
The position vector of \(Q\) is that of \(P\) plus \(\overrightarrow{PQ}\):
\[ \begin{aligned} Q &= (2 + 3,\ 1 - 1,\ -3 + 5) = (5, 0, 2) \\ OQ &= \sqrt{25 + 0 + 4} = \sqrt{29} \end{aligned} \]
Subtracting \(\overrightarrow{PQ}\) from \(P\) instead gives \((-1, 2, -8)\) and a distance of \(\sqrt{69}\), one of the distractors.
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