I have the following equation, with 5 $\mathbb{R}^3$ vectors A, B, C, D, and P, and a scalar, $t$;
$-At^3 + 3Bt^3 + Ct^3 - 3Dt^3 + 3At^2 - 6Bt^2 + 3Dt^2 - 3At + 3Bt + A - P$
I'm trying to find the minimum of this equation's absolute magnitude, but I'm neither sure how to apply the pythagorean formula to a function, nor am I sure how to find the minima and maxima of a vector valued function. If anyone could provide some advice, it would be much appreciated.
Essentially you have four vectors $\vec{v_k},\, k=0,3$ and the vector-valued function
$$ \vec{f}(t)=\sum_{k=0}^3\vec{v}_kt^{k} $$
and you would like to know how close the path of this function gets to $\vec{0}$.
So you should minimize $\vert\vec{f}(t)\vert^2$.
If we let $\vec{v}_k=(x_k,y_k,z_k)$ then
$$ \vert\vec{f}(t)\vert^2 =\left(\sum_{k=0}^3x_kt^{k}\right)^2+\left(\sum_{k=0}^3y_kt^{k}\right)^2+\left(\sum_{k=0}^3z_kt^{k}\right)^2=g(t)$$ This is a sixth degree non-negative polynomial so we know it has a minimum value. But to find it we must solve the fifth degree polynomial equation $g^\prime(t)=0$.
\begin{eqnarray}\frac{1}{2}g^\prime(t)&=&\left(\sum_{k=1}^3kx_kt^{k-1}\right)\left(\sum_{k=0}^3x_kt^{k}\right)\\ &+&\left(\sum_{k=1}^3ky_kt^{k-1}\right)\left(\sum_{k=0}^3y_kt^{k}\right)\\ &+&\left(\sum_{k=1}^3kz_kt^{k-1}\right)\left(\sum_{k=0}^3z_kt^{k}\right)\\ &=&0 \end{eqnarray}
As you know, this fifth-degree polynomial equation cannot be solved analytically so you will have to rely on numerical methods.