Vectors a, b, c are non-zero and satisfy (a x b) x c = (1/3)|b||c||a|. If the angle between vectors b and c is theta, then sin(theta) equals:

1/3
sqrt(2)/3
2/3
2*sqrt(2)/3

Correct answer: 2*sqrt(2)/3

Solution

Applying BAC-CAB: (a x b) x c = (a.c)b - (b.c)a. The right-hand side magnitude is (1/3)|b||c||a|. For consistency with a scalar right side, assume a is perpendicular to c (a.c = 0). Then -(b.c)a = (1/3)|b||c||a|, giving |b.c| = (1/3)|b||c|, i.e. |cos(theta)| = 1/3. Therefore sin(theta) = sqrt(1 - 1/9) = sqrt(8/9) = 2*sqrt(2)/3.

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