Triple scalar product/coplanarity of

[Bipolarity]

Homework Statement

Suppose that a,b,c are nonparallel nonzero vectors, and that $( a \times b) \cdot c = 0$. 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, $P = (t_{1}a + t_{2}b: t_{1},t_{2} \epsilon ℝ)$ 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

 

[HallsofIvy]

So you have to do this the hard way? Then let a= <a, b, c>, b= <p, q, r>, and c= <x, y, z>. Calculate the coordinates of $(a\times b)\cdot c$ and set them equal to 0.

 

[Bipolarity]

So you have to do this the hard way? Then let a= <a, b, c>, b= <p, q, r>, and c= <x, y, z>. Calculate the coordinates of $(a\times b)\cdot c$ and set them equal to 0.

Thanks Ivy. Is there any faster way around it, that involves using vector identities and the like?

BiP