Suppose that an object hangs from three ropes which point in the directions of the vectors ⟨2,−1,2⟩, ⟨−2,2,1⟩, and ⟨−3,0,4⟩. (Assume that the positive z-axis points directly upward.) If the tension in the first of these three ropes is 360 pounds, determine the weight of the object
I assumed the tension given is for the first vector and found its magnitude, so the "scale" is 120. Given this, I found the magnitude for the other vectors.
Now, I assume the fourth force has to cancel out with the other three so the sum of the 4 vectors has to be 0 right? This is as far as I got
This problem is about equilibrium of the forces. You have four forces: three tensions in the ropes, $T_1=360$, $T_2$, and $T_3$, and the weight $W$. So with three unknowns you need to write three equations. In this case, you can write equilibrium along each of the $x, y, z$ axes. $W$ has component only along $z$. So you need to write the components of the tensions in terms of the magnitude (known or unknown) and direction. Use the scalar (or dot) product to find these components.