Trying to solve 3x3 determinant with two zeros

[togo]

Homework Statement

3 wires and unknown tensions: Tab, Tbc, Tbd
Flowerpot being suspended in equilibrium

Fx =  .05Tab - 0.728Tbc - 0.728Tbd = 0 lbs
Fy = -.05Tab + 0.485Tbc + 0.485Tbd = 20 lbs (weight of flower pot)
Fz =    0Tab + 0.485Tbc - 0.485Tbd = 0 lbs

Homework Equations

Is it possible to find tension in one of these wires?

The Attempt at a Solution

Using Cramers rule, came up with

-14.1232/0.117855

which was incorrect.

 

[vela]

You seem to have made an arithmetic error somewhere. Your answer for T_ab is off by a factor of 10.

 

[togo]

I went over it again today and got the same answer :(
back of the book lists 17 lbs for Tab. Here is the determinant I used,

resulting in

$$\frac{-7.0616-7.0616}{(-0.1176125)+(0.17654)-(-0.017654)-(0.1176125)}$$

= 119.83 lbs, Tab

 

[vela]

Are the coefficients of T_ab equal to 0.5 (third post) or 0.05 (original post)? That's the factor of 10 difference between our answers. So the good news is you're basically calculating T_ab correctly using Cramer's rule. The bad news is, if T_ab is supposed to be 17 lb, your equations are wrong.

 

[togo]

0.5, sorry. Not sure where the equations may have gone wrong.

 

[vela]

Just a guess, but I'd expect the coefficients of T_ab should be either $\pm 1/\sqrt{2}$ or one of them should be $\pm \sqrt{3}/{2}$.

More generally, if you form a vector from the coefficients of T_ab, the norm of the vector should be 1.

 

[togo]

I went over the numbers with someone else in my class who also verifies that the coordinates used are correct. It sounds like this problem can be solved using a graphing calculator, but its strange that I was unable to solve it using cramers rule.

 

[vela]

It's not strange that Cramer's rule doesn't work. What's strange is you concluding Cramer's rule only works some of the time rather than concluding your equations are wrong or that you are making some other mistake. If a proved result, like Cramer's rule, doesn't seem to be working, that's an indication that there's something else is wrong.

I can tell you that if you and your classmate got exactly the same equations, you and your classmate are wrong. If your classmate claims to have gotten the right answer from those equations, he or she is making yet another mistake.

I can read off the components of T_ab from your equations. According to them, the force T_ab is equal to$$\vec{T}_{ab} = \frac{T_{ab}}{2} \mathbf{\hat{i}} + \frac{T_{ab}}{2} \mathbf{\hat{j}}$$where T_ab=|T_ab|. But this implies that$$\lvert \vec{T}_{ab} \rvert = T_{ab} \sqrt{\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^2} = T_{ab}\frac{\sqrt{2}}{2} \ne T_{ab}$$which is a contradiction, which in turn indicates your equations are wrong.