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Solving the cross product equation is a well known problem. You can prove that if $\mathbf P=\mathbf Q\times \mathbf X$ (and $\mathbf P\cdot \mathbf Q =0)$ for vectors $\mathbf P,\mathbf Q$ and unknown vector $\mathbf X$, then you have that $$\mathbf X=\frac{\mathbf P\times \mathbf Q}{|\mathbf Q|}+k \mathbf Q$$ where $k$ is a scalar that has to be determined from another equation. You are right to have a zero determinant, you can see that the solution above works for any value of $k$.

The equations of ideal magnetohydrodynamics (MHD) are actually more than 2, you are probably missing other common fluid equations like:

1. the conservation of mass $\partial_t\rho+\nabla\cdot(\rho \mathbf V)$
2. the conservation of momentum $\partial_t (\rho \mathbf V)+\nabla\cdot (\rho \mathbf V \otimes \mathbf V)=-\nabla P +\text{other force terms}$
3. An equation for the energy or for pressure $P$

You will also need boundary/initial conditions to completely determine your system.

Note also that the equation $\mathbf E\cdot \mathbf B=0$ is not always true in ideal MHD, so you should check the hypotheses you are considering.

Solving the cross product equation is a well known problem. You can prove that if $\mathbf P=\mathbf Q\times \mathbf X$ (and $\mathbf P\cdot \mathbf Q =0)$ for vectors $\mathbf P,\mathbf Q$ and unknown vector $\mathbf X$, then you have that $$\mathbf X=\frac{\mathbf P\times \mathbf Q}{|\mathbf Q|}+k \mathbf Q$$ where $k$ is a scalar that has to be determined from another equation. You are right to have a zero determinant, you can see that the solution above works for any value of $k$.

The equations of ideal magnetohydrodynamics (MHD) are actually more than 2, you are probably missing other common fluid equations like:

1. the conservation of mass $\partial_t\rho+\nabla\cdot(\rho \mathbf V)=0$
2. the conservation of momentum $\partial_t (\rho \mathbf V)+\nabla\cdot (\rho \mathbf V \otimes \mathbf V)=-\nabla P +\text{other force terms}$
3. An equation for the energy or for pressure $P$

You will also need boundary/initial conditions and fluid properties to completely determine your system.

Note also that the equation $\mathbf E\cdot \mathbf B=0$ is not always true in ideal MHD, so you should check the hypotheses you are considering.

Solving the cross product equation is a well known problem. You can prove that if $\mathbf P=\mathbf Q\times \mathbf X$ (and $\mathbf P\cdot \mathbf Q =0)$ for vectors $\mathbf P,\mathbf Q$ and unknown vector $\mathbf X$, then you have that $$\mathbf X=\frac{\mathbf P\times \mathbf Q}{|\mathbf Q|}+k \mathbf Q$$ where $k$ is a scalar that has to be determined from another equation. You are right to have a zero determinant, you can see that the solution above works for any value of $k$.

The equations of ideal magnetohydrodynamics are actually more than 2, you are probably missing other common fluid equations like:

1. the conservation of mass $\partial_t\rho+\nabla\cdot(\rho \mathbf V)$
2. the conservation of momentum $\partial_t (\rho \mathbf V)+\nabla\cdot (\rho \mathbf V \otimes \mathbf V)=-\nabla P +\text{other force terms}$

Solving the cross product equation is a well known problem. You can prove that if $\mathbf P=\mathbf Q\times \mathbf X$ (and $\mathbf P\cdot \mathbf Q =0)$ for vectors $\mathbf P,\mathbf Q$ and unknown vector $\mathbf X$, then you have that $$\mathbf X=\frac{\mathbf P\times \mathbf Q}{|\mathbf Q|}+k \mathbf Q$$ where $k$ is a scalar that has to be determined from another equation. You are right to have a zero determinant, you can see that the solution above works for any value of $k$.

The equations of ideal magnetohydrodynamics (MHD) are actually more than 2, you are probably missing other common fluid equations like:

1. the conservation of mass $\partial_t\rho+\nabla\cdot(\rho \mathbf V)$
2. the conservation of momentum $\partial_t (\rho \mathbf V)+\nabla\cdot (\rho \mathbf V \otimes \mathbf V)=-\nabla P +\text{other force terms}$
3. An equation for the energy or for pressure $P$

You will also need boundary/initial conditions to completely determine your system.

Note also that the equation $\mathbf E\cdot \mathbf B=0$ is not always true in ideal MHD, so you should check the hypotheses you are considering.

Solving the cross product equation is a well known problem. You can prove that if $\mathbf P=\mathbf Q\times \mathbf X$ (and $\mathbf P\cdot \mathbf Q =0)$ for vectors $\mathbf P,\mathbf Q$ and unknown vector $\mathbf X$, then you have that $$\mathbf X=\frac{\mathbf P\times \mathbf Q}{|\mathbf Q|}+k \mathbf Q$$ where $k$ is a scalar that has to be determined from another equation. You are right to have a zero determinant, you can see that the solution above works for any value of $k$. You will need equations from fluid dynamics or initial/boundary conditions to completely determine $\mathbf V$.

Solving the cross product equation is a well known problem. You can prove that if $\mathbf P=\mathbf Q\times \mathbf X$ (and $\mathbf P\cdot \mathbf Q =0)$ for vectors $\mathbf P,\mathbf Q$ and unknown vector $\mathbf X$, then you have that $$\mathbf X=\frac{\mathbf P\times \mathbf Q}{|\mathbf Q|}+k \mathbf Q$$ where $k$ is a scalar that has to be determined from another equation. You are right to have a zero determinant, you can see that the solution above works for any value of $k$.

The equations of ideal magnetohydrodynamics are actually more than 2, you are probably missing other common fluid equations like:

1. the conservation of mass $\partial_t\rho+\nabla\cdot(\rho \mathbf V)$
2. the conservation of momentum $\partial_t (\rho \mathbf V)+\nabla\cdot (\rho \mathbf V \otimes \mathbf V)=-\nabla P +\text{other force terms}$