# Math Help - Equilibrium Independence

1. ## Equilibrium Independence

Can someone show that the static equilibrium equations in two and three dimensions on any body are linearly independent.

2. Originally Posted by ThePerfectHacker
Can someone show that the static equilibrium equations in two and three dimensions on any body are linearly independent.
I am unable to understand your question.

Keep SMiling
Malay

3. Originally Posted by malaygoel
I am unable to understand your question.
It is a linear algebra question.

Note the equations,
$\left\{ \begin{array}{cc}x+y=2 \\ 2x+2y=4 \end{array} \right\}$
You have 2 equations, 2 varaibles.
But no unique solution?
Why?
Because they are not linearly independent:
Meaning, linear dependent:

Let, the vectors,
$\bold{x}_1=<1,1>$
$\bold{x}_2=<2,2>$
The coefficients of the system.

Then we want to know "can be find non-zero (non-trivial) solution" to:
$k_1\bold{x}_1+k_2\bold{x}_2=\bold{0}$
Of couse, $k_1=k_2=0$.
But I ask different solutions (non-trivial).
And we can,
$2<1,1>-<2,2>=<0,0>$.
That tells us that the system has no unique solution.

Another way, is through the determinant,
$\left| \begin{array}{cc} 1&1\\2&2 \end{array} \right|=0$
Since it is zero there cannot be a unique solution.
----
Thus, I am asking that the static equilbrium equation provide us with enough equations for unknowns. But how do we know it solves?

4. Originally Posted by ThePerfectHacker
It is a linear algebra question.

Note the equations,
$\left\{ \begin{array}{cc}x+y=2 \\ 2x+2y=4 \end{array} \right\}$
You have 2 equations, 2 varaibles.
But no unique solution?
Why?
Because they are not linearly independent:
Meaning, linear dependent:

Let, the vectors,
$\bold{x}_1=<1,1>$
$\bold{x}_2=<2,2>$
The coefficients of the system.

Then we want to know "can be find non-zero (non-trivial) solution" to:
$k_1\bold{x}_1+k_2\bold{x}_2=\bold{0}$
Of couse, $k_1=k_2=0$.
But I ask different solutions (non-trivial).
And we can,
$2<1,1>-<2,2>=<0,0>$.
That tells us that the system has no unique solution.

Another way, is through the determinant,
$\left| \begin{array}{cc} 1&1\\2&2 \end{array} \right|=0$
Since it is zero there cannot be a unique solution.
----
Thus, I am asking that the static equilbrium equation provide us with enough equations for unknowns. But how do we know it solves?
Yes we know what what linear independence means, the obscurity is in
what you mean by the static equilibrium equations.

For static equilibrium we require that the sum for the foces on a body/system
is zero, as is the sum of the torques on the body/system.

RonL

RonL

5. Originally Posted by ThePerfectHacker
Can someone show that the static equilibrium equations in two and three dimensions on any body are linearly independent.