Solving a over determined system in MATLAB

I have 4 equations and 3 variables. I know that 3 equations are sufficient but my instructor said that the equations make a overdetermined system. My equations are:

e1=(x*(a-principle_stress(1,1)))+(p*y)+(r*z);
e2=(y*(b-principle_stress(1,1)))+(p*x)+(q*z);
e3=(r*x)+(q*y)+(z*(c-principle_stress(1,1)));
eqn4 = (x^2) + (y^2) + (z^2) == 1;

What I did earlier was

eqns=[e1,e2,eqn4];
S = solve(eqns,[x y z]);

The above method matched with the actual answer but is it the correct way of solving a overdetermined system?

Will using

eqns=[e1,e2,e3,eqn4]

solve my problem?

Answer 1

[John D'Errico]

Is it the correct way? perhaps it might be, except that you have a typo in e3. Perhaps that is why you decided to try it with only 3 equations. Look carefully at e3. Does it contain y at all? (NO.) TYPO.

It looks like you are trying to solve an eigenvector problem, thus you have an eigenvalue of a 3x3 stress matrix, well, you call it a principle stress. It is still one of three eigenvalues of that 3x3 matrix.

How should you solve the problem? Best is just to use eig, which will give you both the desired stresses and the eigenvectors directly. Trying to use solve at all here is a bad idea, because it forces you to face a numerical problem.

If you already know the value of the principle stress, then you can use null to give the desired vector directly.

Of course, since this is homework, I assume you are not allowed to use either null or eig anyway.

Sigh.

[Kyathari Hrushikesh]

Thanks for your reply, I have actually used null to solve for the variables but by doing that, I am not using the 4th equation in any way to obtain the solutions. The principle stress values are known but there is actually no way for me to prove that even the 4th equation is satisfied by the values that I obtain using null(unless i use if after getting the results).

[John D'Errico]

I'm not sure what is your problem. Part of it is you apparently do not understand null, since it already returns a result that is normalized to have unit norm, thus implicitly satisfying the 4th equation.

So, why do you feel the need to "prove" the 4th equation is satisfied by the result of null?

I'd suggest you would be further ahead removing what appear to be the typo from your code. If you try to solve the correct equations, using the correct tools (here, null) then you will get the necessary result.

Finally, I would point out that symbolic tools are entirely overkill in all of this.

[Kyathari Hrushikesh]

I didn't knew that null returns a normalized unit norm, thanks for your time and patience