If each of the vectors $u, v, w$ in $\Bbb R^3$ is perpendicular to the other two, then the three vectors are linearly independent.
When is this true/ false?
UPDATE: A simple (such that a first year student can understand) solution can be done like this:
By observing that the set is not linearly independent if $u,v,w=0$
If the vectors are orthogonal (pairwise), then: $$u\cdot v=0$$ $$u\cdot w=0$$ and $$v \cdot w=0$$
If $u,v,w$ are linearly independent, then the only solution for: $$c_1u + c_2v + c_3w=0 \ \ \ (1)$$ is $$c_1=c_2=c_3=0$$
Multiply $(1)$ by $u \rightarrow$ $$c_1u^2+c_2(u\cdot v) + c_3(u\cdot w)=0\cdot u$$ and so $c_1=0$ since $u\cdot w=0$ and $u\cdot v=0$ (as they are orthogonal).
We can repeat the same procedure, and we get that the only solution to $(1)$ is trivial, hence they are linearly independent.