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Bipolarity
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Homework Statement
Suppose that a,b,c are nonparallel nonzero vectors, and that [itex] ( a \times b) \cdot c = 0 [/itex]. Show that c is expressible as a linear combination of a and b. Avoid geometric arguments (that is, try to stick to vector algebra and symbols in the proof).
Homework Equations
The Attempt at a Solution
The geometric interpretation is that because the triple scalar product of a,b,c is 0, the three vectors are coplanar. Thus, c lies on the plane P determined by a,b. In other words, c is an element of the set of vectors given by the parametrization of the plane P, namely, [itex] P = (t_{1}a + t_{2}b: t_{1},t_{2} \epsilon ℝ) [/itex] And thus, c is expressible as a linear combination of a and b.
But I'm trying to prove this algebraically using vector identities, matrices, and the like. No geometry. Any ideas? I appreciate all help thanks!
BiP
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