I have two simple vector equations:

$\displaystyle \vec{F_{T}}=\sum_{i=0}^n{\vec{F_{i}}}$

and:

$\displaystyle \vec{L_{T}}=\sum_{i=0}^n{\vec{F_{i}} \times \vec{H_{i}}}$ : ( $\displaystyle \times$ being the vector (aka cross) product).

I know $\displaystyle \vec{F_{T}}$ , $\displaystyle \vec{L_{T}}$ and all values of $\displaystyle \vec{H_{i}}$. I want to findsensiblevalues for the $\displaystyle \vec{F_{i}}$ .

When n = 2 this is a simple simultaneous equation.

However when n > 2 there is no single point solution.

Consider n=3; one could view the two equations as defining planes, the intersection of these planes would then define a line of points that satisfy the two equations.

What I want is the solution from this range that minimises the largest value of $\displaystyle \vec{F_{i}}$.

My current thinking is set all except 2 of the $\displaystyle \vec{F_{i}}$ to zero and solve the resulting simultaneous equation.

Repeat for a different pair of $\displaystyle \vec{F_{i}}$

This gives two points on the solution line.

Find the perpendicular to this line that passes through the origin (All $\displaystyle \vec{F_{i}}=0$)

Am I looking in the correct area? Can someone point the way forward please.