1
$\begingroup$

I read that three vectors in $\mathbb R^3$ are coplanar if one is a linear combination of the other two. I tried to prove this statement but couldn't do it.

I know that three vectors $\mathbf u,\,\mathbf v,\,\mathbf w \in \mathbb R^3$ are complanar if and only if $$\mathbf u \cdot (\mathbf v \times \mathbf w) = 0.$$

Can I use this fact to prove the above statement? Also is the statement an "if" or an "if and only if"?

$\endgroup$
2
$\begingroup$

Suppose $\;\mathbf u=a\mathbf v+b\mathbf w\;, \;\;a,b\in\Bbb R\;$ , then using the properties of cross product:

$$\mathbf u\cdot(\mathbf v\times\mathbf w)=(a\mathbf v+b\mathbf w)\cdot(\mathbf v\times \mathbf w)=a\mathbf v\cdot(\mathbf v\times\mathbf w)+b\mathbf w\cdot(\mathbf v\times\mathbf w)=0$$

since $\;\mathbf v\times\mathbf w\;$ is a vector perpendicular to both $\;\mathbf v\,,\,\,\mathbf w\;$ .

The iff statement enters only when $\;\mathbf v\,,\,\mathbf w\neq\mathbf 0\;$ , otherwise we get no plane at all.

$\endgroup$
  • $\begingroup$ Thanks! Your proof is really clear. I see you wrote $\mathbf 0$ instead of $0$ when comparing to vectors. That's what I write as well in my notes. My book instead keeps writing $0$ even in (in)equalities with vectors, but I think it's wrong, isn't it? The book only defined equality between vectors. $\endgroup$ – rubik Mar 8 '15 at 8:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.