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Let \(\vec p, \vec q, \vec r\) be three pairwise perpendicular vectors, each having the same magnitude. If a vector \(\vec x\) satisfies \[ \vec p\times[(\vec x-\vec q)\times\vec p]+\vec q\times[(\vec x-\vec r)\times\vec q]+\vec r\times[(\vec x-\vec p)\times\vec r]=\vec 0, \] then \(\vec x\) is

1. \(\tfrac12(\vec p+\vec q-2\vec r)\)
2. \(\tfrac12(\vec p+\vec q+\vec r)\)
3. \(\tfrac13(\vec p+\vec q+\vec r)\)
4. \(\tfrac13(2\vec p+\vec q-\vec r)\)

Correct answer: \(\tfrac13(\vec p+\vec q+\vec r)\)

Solution

The correct option is \\( frac{1}{3}( extbf{p} + extbf{q} + extbf{r})\\ because it represents the centroid of the triangle formed by the three pairwise perpendicular vectors, ensuring that the contributions from each vector balance out to satisfy the given vector equation.

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