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Let a, b, and c represent three unit vectors that are not in the same plane, with the angle between each pair being π/3. If the expression a × b + b × c equals pa + qb + rc, where p, q, and r are constants, what is the value of (p² + 2q² + r²)/q²?
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- 4
Correct answer: 4
Solution
With a.b=b.c=c.a=1/2, dotting a x b + b x c = pa+qb+rc by a,b,c gives p+q/2+r/2=V, p/2+q+r/2=0, p/2+q/2+r=V where V=[a b c]. Solving: p=r=V and q=-V, so (p^2+2q^2+r^2)/q^2 = (V^2+2V^2+V^2)/V^2 = 4. Correct option is index 3; stored index 0 is wrong.
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