Determine the unit vectors for each force's direction.

Step 1: Determine the unit vectors for each force's direction.
The given direction vectors are $\vec{d_1} = (i+3j)$, $\vec{d_2} = (2i-\sqrt{3}j)$, and $\vec{d_3} = (i+\sqrt{5}j)$.
The magnitude of each direction vector is:
$|\vec{d_1}| = \sqrt{1^2 + 3^2} = \sqrt{1+9} = \sqrt{10}$
$|\vec{d_2}| = \sqrt{2^2 + (-\sqrt{3})^2} = \sqrt{4+3} = \sqrt{7}$
$|\vec{d_3