How to maximize the dot product of two vectors, one is fixed, the other is constrained on an ellipse?
i.e., how to maximize $$ v_0 x+ v_1 y $$ s.t. $$ \left(\frac{x}{a} \right)^2 +\left(\frac{y}{b} \right)^2=1 $$
intuitively, let $x=a \sin t, y= b \cos t$, then the limit occurs when the tangent $$ \begin{bmatrix} a \cos t \\ -b \sin t \\ \end{bmatrix} $$ is orthogonal to $$ \begin{bmatrix} v_0 \\ v_1 \end{bmatrix} $$ i.e. $$ v_0 a \cos t= v_1 b \sin t $$ Thus, the maximum point is $$ \begin{cases} x=\frac{a^2 v_0}{ \sqrt{ a^2 v_0^2 +b^2 v_1^2 }} \\ y=\frac{b^2 v_1}{ \sqrt{ a^2 v_0^2 +b^2 v_1^2 }} \end{cases} $$
Without multipliers:
Let $x=a\cos t,y=b\sin t$ and maximize
$$v_0a\cos t+v_1b\sin t=\sqrt{(v_0a)^2+(v_1b)^2}\cos(t-\phi).$$
Obviously, the maximum is
$$\sqrt{(v_0a)^2+(v_1b)^2}.$$