Let $p,v,t$ be the position, velocity and time unknowns such that for all $i$ we have a collision
$$p-p_i - t\times(v-v_i)=0.$$
To eliminate $t$ we wedge (exterior) product with $v-v_i$ to get
$$0=(p-p_i)\wedge (v-v_i)= p\wedge v - p \wedge v_i - p_i \wedge v + p_i \wedge v_i.$$
The nonlinear term $p\wedge v$ is constant for all $i$, so we subtract equations to get a linear equation
$$p \wedge (v_i-v_j) + (p_i - p_j) \wedge v - p_i \wedge v_i + p_j \wedge v_j=0.$$
Then wedge with $p_i - p_j$ to eliminate the $v$ unknowns
$$p \wedge (v_i-v_j) \wedge (p_i - p_j) + p_i \wedge v_i \wedge p_j + p_j \wedge v_j \wedge p_i =0.$$
This is an equation with three unknowns so pick three combinations for $i,j$ and solve the system
@abroy77 I just mean multiplication, it's given the same symbol as the cross product.
Ambivalence like this is an antipattern since you have to know the arguments to deduce the operator.