Given two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on a finite-dimensional vector space, could it be that there exist vectors $\vec{v}_1,\vec{v}_2$ such that $$\|\vec{v}_1\|_a>\|\vec{v}_2\|_a$$ and $$\|\vec{v}_1\|_b<\|\vec{v}_2\|_b$$ i.e. that the norms "disagree" on which vector is the longest?
Maybe this is related to norm equivalence, but I don't quite see how to get here from there.
On
Sure. An example in $\Bbb{R}^2$: \begin{align} ||(x,y)||_a &= \sqrt{\left( \frac{x}{2} \right)^2 + y^2} \\ ||(x,y)||_b &= \sqrt{x^2 + \left( \frac{y}{2} \right)^2} \end{align} The sets of constant norm under these two norms are ellipses, but the major axes with the $a$ norm are along the $x$-axis and the major axes for the $b$-norm are long the $y$-axis. This suggests looking at \begin{align} ||(0,1)||_a &= 1 & ||(1,0)||_a &= 1/2 \\ ||(0,1)||_b &= 1/2 & ||(1,0)||_b &= 1 \text{.} \end{align}
Yes. Otherwise the norms would only be multiples of each other.
$$\|(30,40)\|_2=50 > 45 =\|(0,45)\|_2$$ $$\|(30,40)\|_\infty=40 < 45 =\|(0,45)\|_\infty$$