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Let a, b, and c be three vectors that are not in the same plane. Define p = (b × c)/([abc]), q = (c × a)/([abc]), and r = (a × b)/([abc]). Then the value of (a+b)·p + (b+c)·q + (c+a)·r is

1. 0
2. 1
3. 2
4. 3

Correct answer: 3

Solution

p,q,r form the reciprocal basis, so a.p=b.q=c.r=1 and all cross terms (b.p, c.q, a.r, etc.) = 0. Thus (a+b).p = 1, (b+c).q = 1, (c+a).r = 1, giving total = 3.

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