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

1. (1)/(2)(p+q-2r)
2. (1)/(2)(p+q+r)
3. (1)/(3)(p+q+r)
4. (1)/(3)(2p+q-r)

Correct answer: (1)/(2)(p+q+r)

Solution

With p,q,r mutually perpendicular and equal magnitude, each triple product term p x [(x-q) x p] = |p|^2(x-q)-(p.(x-q))p. Summing all three and setting equal to 0 yields x = (1/2)(p+q+r).

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