Simple problem with vectors in 3 dimensions

[Wolftacular]

Homework Statement

Three cables are used to tie the balloon shown in the figure. Determine the vertical component of the force P exerted over the balloon at point A, if the tension on cable AB is 259 N.

http://img521.imageshack.us/img521/9687/physicsballoonproblem.png

(Sorry for the figure. I had to re-draw it myself, on paint, with a touch pad. Assume all measurements are parallel to the axises, of course)

Homework Equations

Pythagorean theorem.

The Attempt at a Solution

Figured out the resulting distance of AB with Pythagorean theorem and then used cross multiplying to find out the equivalent vertical component of AB. Then...

What I did: Simply multiplied that resulting vertical force (207.2 N) times three for each of the cables, for a vertical resultant of 621.6 N.

What the teacher did: He also found the x component of AB and equaled it to the x component of CA, found the z component and equaled it to that of AD. Then, with lots of triangle solving, found the three different y components and added them all up for a result of 1000-something N.

MY QUESTION:
How would you go about solving this problem? The teacher reasoned that the x and z components had to be in equilibrium to find the y resultant. I disagree because he never stated that the balloon was in equilibrium anywhere. Is he wrong? Or should I always assume that, when finding a component, all other components should be balanced?

This is a test question and he's very stubborn so I want to know what you all think.

 

[voko]

"Tie the balloon" implies that the cable and the balloon are stationary - or at least can become stationary if released. And when the system is stationary, it is in equilibrium.

You could also consider that, geometrically, three cables can meet, when fully extended, at two points at most (can you see why?), so it has to be stationary at one of those points.

 

[phinds]

What I did: Simply multiplied that resulting vertical force (207.2 N) times three for each of the cables, for a vertical resultant of 621.6 N.

Uh ... you really need to think that through. How can you possibly justify that? Perhaps thinking about that will help you see what actually has to be done.

 

[Wolftacular]

So I suppose my mistake (as well as the rest of my classmates') was to assume that the graphical vector representation of the distances (on the picture) was identical to that of the forces, when in reality, it's not. Hence why the vertical distances might be the same but not the tension in these three vertical components. Right?

 

[phinds]

So I suppose my mistake (as well as the rest of my classmates') was to assume that the graphical vector representation of the distances (on the picture) was identical to that of the forces, when in reality, it's not. Hence why the vertical distances might be the same but not the tension in these three vertical components. Right?

Well think about it this way: If you push a box in a North West direction, is all of the force you expend going towards moving the box in the Northernly direction?

Do you understand how to use force vectors?

 

[Wolftacular]

Well think about it this way: If you push a box in a North West direction, is all of the force you expend going towards moving the box in the Northernly direction?

Do you understand how to use force vectors?

Yes I do, and if I'm to take your example as the basis of what you think my mistake really was, I think you're confused. Either way, my doubts have been cleared, so thank you!