(1) If a, b, c are three linearly independent vectors show that the vectors a × b, b × c, c × a are also linearly independent.
(2) For any vector x, we express it in terms of these latter vectors x = λa × b + µb × c + νc × a,
λ, µ, ν ∈ R. Find λ, µ and ν in terms of a, b, c and x.
No idea how to approach this. thanks.
Let $u=a \times b$ and so on. By definition, $u$ is orthogonal to both $a$ and $b$. Now consider, \begin{align*} c_1u+ c_2v +c_3w & = \mathbf{0}\\ (c_1u+ c_2v +c_3w) \cdot a & = \mathbf{0} \cdot a\\ c_2(a \cdot (b \times c)) & = 0. \end{align*} But $a,b,c$ are linearly independent, therefore $a \cdot (b \times c) \neq 0$, otherwise they will coplanar. This means $c_2=0$. Likewise we can get that all coefficients are $0$, thus linear independence.
Now use the same idea of taking dot product with appropriate vectors to solve for part (2).