Here are two n-dimensional vectors: $V_1$ and $V_2$
$V_1 (v_1,v_2, \dots ,v_n)$
$V_2 (v_1,v_2, \dots ,v_n)$
It seems that $V_1*cos(\theta) + V_2*sin(\theta)$ is an ellipse in the n-D space. (Its center is the origin.)
Here is another point $P (p_1,p_2, \dots ,p_n)$
I want to find a point E on the ellipse that has the shortest euclidean distance from P.
My current idea is to first project the point onto the plane that is spanned by $V_1$ and $V_2$ and then try to solve the problem in 2-D space.
Maybe I can find $a$ and $b$ such that P = $a*V_1+ b*V_2$ and then solve the nearest-point problem in the $V_1$ and $V_2$ 2-D subspace. But it seems problematic to get the 2-D ellipse equation since $V_1$ and $V_2$ are not necessarily orthogonal.
What's the direction and length of the major and minor axes of the ellipse: $V_1*cos(\theta) + V_2*sin(\theta)$ ?