A force $F_1 = i - 3j -2k$ at the point $r_1 = -2i + 9j$ another force $F_2 = 2i + j -3k$ at the point $r_2 = i + yj -k$ and a third force $F_3$ are equivalent to zero. Find y for this to be possible. Find $F_3$ and its line of action in this case
Well, I've found y to be 13 by finding $F_3$ by $\sum_{i=1}^3 F_i = $ and found y by $\sum_{i=1}^n r_i \times F_i = 0$, but I'm having troubles with the last part
in all honesty I don't know what's going on, I found y = 13 by simply guessing as we done something similar in lecture, I don't UNDERSTAND what's happening, not at all. If someone could help with the last part and any explanation on what's going on would be great
Pertaining to the last part of the question and the comments:
In this situation, the coordinate system you are using models the position vectors of your body. The forces however, are not bound by this origin, and they are different depending on where they act. A force $(2, 1, 3)$ acting on, say, $(1, 1, 1)$ will be objectively different than the 'same' vector acting on $(0,1,3)$ i.e. it might make the body rotate or just push it forward.
The mathematical formulation of this is that position vectors lie in a vector space, whereas force vectors are in a affine space (wikipedia for more). In a vector space every vector is specified by its coordinates, whereas in an affine space there is no origin, and the vectors are determined by their end points.
Specifically within this context, if you solve for $F_3$ plugging in all the conditions, you will see it can lie in a certain vector line, that varies according to a parameter. The standard derivation of this fact involves solving the system
$\bf a \wedge x = \bf b \\ x = \alpha \,a + \beta \, b + \gamma \, (a\wedge b)$
where the solution turns out to be a line.
Even if you found out that it acted on a specific point, the action line of the force wouldn't be the direction of the force vector from the origin, but the line $r_3 + \lambda F_3$.
Also, I believe we're in the same class, so feel free to come talk tomorrow if you have any further questions :).