Clearly, any parameter-family which is not contained in the line will contain such a vector (this is just rephrasing the condition of being not parallel). Therefore, if we manage to find a one-parameter family which is not contained in any line through the origin then this construction will work for arbitrary vectors .

The simplest such construction is to simply fix two of the three components of your vector with at least one of them being non-zero (i.e., to choose as your one-parameter family a one-dimensional affine space which does not contain the origin), e.g.,

.

Then given a specific vector we only need to choose in such a way that the last two components of are not parallel to those of . Suppose that we had failed to do so and the two -vectors are in fact parallel. Then

.

It follows that we can simply choose

to ensure that the two vectors and are non-parallel (one quickly verifies that this also works for as long as one defines ).

$\frac{\partial \phi}{\partial t} = \nabla \cdot \vec{A}$

$\vec{A} \times \vec{B} = \vec{C}$ and $\frac{\partial \phi}{\partial t} = \nabla \cdot \vec{A}$

and $\frac{\partial \phi}{\partial t} = \nabla \cdot \vec{A}$

Maxwell’s equations:

<br>
<br>
<br>
<br>

Maxwell’s equations:

$latex[\vec{\nabla} \cdot \vec{\mathbf{E}} = \frac{1}{\epsilon_0}\,\rho ]$ <br>
$latex[\vec{\nabla} \times \vec{\mathbf{E}} = -\frac{\partial \vec{\mathbf{B}}{\partial t} ]$ <br>
$latex[\vec{\nabla} \cdot \vec{\mathbf{B}} = 0 ]$ <br>
$latex[\vec{\nabla} \times \vec{\mathbf{B}} = \mu_0\vec{mathbf{J}} + \epsilon_0\frac{\partial \vec{\mathbf{E}}{\partial t} ]$ <br>

$latex[\vec{\nabla} \cdot \vec{\mathbf{E}} = \frac{1}{\epsilon_0}\,\rho ]$
$latex[\vec{\nabla} \times \vec{\mathbf{E}} = -\frac{\partial \vec{\mathbf{B}}{\partial t} ]$
$latex[\vec{\nabla} \cdot \vec{\mathbf{B}} = 0 ]$
$latex[\vec{\nabla} \times \vec{\mathbf{B}} = \mu_0\vec{mathbf{J}} + \epsilon_0\frac{\partial \vec{\mathbf{E}}{\partial t} ]$