an easy question that I completely understand, just not sure how to algebraically prove.
$u,v,w$ are vectors in $R^3$
given $u\times v +v\times w + w\times u=0$
I need to prove that {u,v,w} are linearly dependent.
Would appreciate your help.
an easy question that I completely understand, just not sure how to algebraically prove.
$u,v,w$ are vectors in $R^3$
given $u\times v +v\times w + w\times u=0$
I need to prove that {u,v,w} are linearly dependent.
Would appreciate your help.
Use the fact that $v\times v=0$: $u\times v +v\times w + w\times u=u\times(v-w)+v\times w-v\times v= (u-v)\times(v-w)=0$ Meaning that: $(u-v)=c(v-w)$ Or that: $u=c(v-w)+v$
Lemma: a nonzero vector $x$ in $\mathbb{R^3}$ can't be perpendicular to three linearly independent vectors $u, v, w$. Proof: three linearly independent vectors make a basis, so we can write $x = a_1u + a_2v + a_3w$. Dotting both sides with $x$ tells you that $x$ is 0.
Main result: Dot both sides of the given equality with $u$. We have $u\cdot (v \times w) = 0$. Thus $v \times w$ is perpendicular to $u, v,$ and $w$.
It follows that either $u, v,$ and $w$ are linearly dependent or $v \times w = 0$, but in this latter case certainly $u, v, w$ are linearly dependent since $v, w$ are.