I have been asked to solve the following problem for my differential geometry class:
Let u(t) = ($u_1$(t), $u_2$(t), $u_3$(t)) and v(t) = ($v_1$(t), $v_2$(t), $v_3$(t)) be differentiable maps from the interval (a,b) into $\mathbb{R}^3$. If the derivatives u'(t) and v'(t) satisfy the conditions: $$u'(t) = au(t) +bv(t) $$ $$ v'(t) = cu(t) - av(t) $$
where a, b, and c are constants, show that u(t) ∧ v(t) is a constant vector.
I am having some issues with this problem. I think part of that comes from the fact that I don't really understand what the constant vector. If anybody could answer that and offer some advice on how to begin that would be great. I guess I also don't really understand why we would need to the derivative condition as well.
Hint: You want to show $\frac{d}{dt}[u(t)\wedge v(t)] = 0$. Consider using the product rule $$\frac{d}{dt}[u(t)\wedge v(t)] = u'(t) \wedge v(t) + u(t) \wedge v'(t)$$ and substituting the expressions of $u'$ and $v'$ into the right-hand side the above equation.