Question
Vectors a, b, and c satisfy a ⋅  ( b × c ) = 0, and
no two of them are zero. Which of the following is necessarily true?Solution
The scalar triple product a • (b × c) = 0 signifies that the volume of the parallelepiped formed by the vectors a, b, and c is zero. A zero volume directly implies that the three vectors lie in the same plane, meaning they are coplanar. Now consider the other options: a ⊥ b is not necessary; vectors can be coplanar without a and b being perpendicular. a ∥ (b × c) would mean a • (b × c) = |a||b × c| ≠0 (unless one of the vectors is zero), so this is also not necessarily true. In fact, a • (b × c) = 0 implies that a is perpendicular to (b × c). b ⊥ c is also not necessary; even if b and c are not perpendicular, the three vectors can still be coplanar.
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