= Solution
If the three <vectors> lie in one <plane> through the origin and $\mathbf u,\mathbf v$ are independent, their <cross product> is a <normal vector> to that <plane>, so its <dot product> with $\mathbf w$ is zero. If $\mathbf u,\mathbf v$ are dependent, their <cross product> is already zero. Thus coplanarity always forces the <scalar triple product> to vanish.
Taking the <contrapositive> proves that a nonzero <scalar triple product> rules out coplanarity. Geometrically its <absolute value> is the nonzero <volume> of the spanned <parallelepiped>.
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