Matrix algebra

[jsbeckton]

I have reduced a physics problem involving a traffic light suspended by 3 cables to the the coordinates of the three points A,B and C with D being the origin.

A=(4,-8,5)
B=(-6,-8,5)
C=(0,8,5)

I am given the weight of the traffic light as 100kg--->981N
Its for a statics class so the idea is that the sum of the forces equal 0
I need to find the tensions on the three lines.

I've reduced the coordinates to unit vectors

\(\displaystyle \begin{array}{l}
\sum _X = \frac{4}{{\sqrt {105} }}T_A \frac{{ - 6}}{{5\sqrt 5 }}T_B 0T_C = 0 \\
\sum _Y = \frac{{ - 8}}{{\sqrt {105} }}T_A \frac{{ - 8}}{{5\sqrt 5 }}T_B \frac{8}{{\sqrt[{}]{{89}}}}T_C = 0 \\
\sum _Z = \frac{5}{{\sqrt {105} }}T_A \frac{5}{{5\sqrt 5 }}T_B \frac{5}{{\sqrt {89} }}T_C = 500 \\
\end{array}\)

Now, I want to know what the best way to solve this would be. I think that it would be really difficult to solve by reducing to rref with these numbers. Any suggestions? My book just kinda works to this point and then gives the anwser so its making me think that I'm missing something obvious. Am I going abouit this the right way?

Thanks!

 

[Unco]

If that is

\(\displaystyle \L
\begin{align*}
\mbox{ \frac{4}{\sqrt{105}}T_A - \frac{6}{5\sqrt{5}}T_B + 0T_c }&= 0\\
\mbox{ -\frac{8}{\sqrt{105}}T_A - \frac{8}{5\sqrt{5}}T_B + \frac{8}{\sqrt{89}}T_C } &= 0 \\
\mbox{ \frac{5}{\sqrt{105}}T_A + \frac{5}{5\sqrt{5}}T_B + \frac{5}{\sqrt{89}}T_C } &= 500\\
\end{align*}\)

You might let

\(\displaystyle \L \mbox{ P = \frac{T_A}{\sqrt{105}}; Q = \frac{T_B}{5\sqrt{5}}; R = \frac{T_C}{\sqrt{89}}\)

So that you have

\(\displaystyle \L
\begin{align*}
\mbox{ 4P - 6Q }&= 0\\
\mbox{ -8P - 8Q + 8R} &= 0 \\
\mbox{ 5P +5Q + 5R} &= 500 \\
\end{align*}\)

 

[jsbeckton]

I'm sorry I made a mistake, the last equation should be set to 981, not 500. I have been going back and forth between my homework problem value (981), and the example value (500). I do not follow how you made that jump. I used the equations solver on my ti-89 to solve that matrix and got the correct anwsers:

TA=603N
TB=439N
TC=925N

How would I go about doing this by hand? I subed 981 for 500 in your matrix and got -

R=490
Q=196.6
P=294.9

Then I took that info and subed it into your equations for P,Q and R to solve for Ta,Tb and Tc but got the wrong anwsers

Ta=3021
Tb=2198
Tc=4623

Can you show me the steps and kinda explain a little? Thanks for all of your help.

 

[Unco]

I was just suggesting a way to make the simultaneous equations a little nicer.

The "jump" is merely recognising the common factor along each column.

If you solve the system of P, Q and R (with 981 instead of 500) you get P = 2943/50 and so forth; so T_A = 2943/50 * sqrt(105) = 603.14 approx, and so forth.

 

[jsbeckton]

I must have mada a matrix algebra mistake, let me go back and check, Thanks.

 

[jsbeckton]

I got it, thanks alot!