= Solution
Three vectors in real three-dimensional space have zero <determinant> exactly when they are <linearly dependent>. Their <linear span> then has dimension at most two, so they lie in a common plane through the origin. Conversely vectors in such a plane span at most two dimensions, so their <scalar triple product> vanishes. This includes parallel nonzero vectors: they need not determine a unique plane, but they lie in at least one. Even zero vectors satisfy the determinant characterization, though the question excludes them.
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