How can I find an equation equal to a span? [closed]

I can't seem to figure out this problem:

Let $$y=\begin{bmatrix}y_1\\y_2\\y_3\end{bmatrix} ,z=\begin{bmatrix}z_1\\z_2\\z_3\end{bmatrix} $$ Find an equation $ax_1 + bx_2 + cx_3 = d$ such that the set of solutions $x=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$ to this equation is equal to span{y, z}.

I have replaced the constants in $v_1$ and $v_2$ with variables to keep the problem anonymous. No y or z constants are zero. Could I get some guidance on how to solve this type of problem? Thanks.

Solution 1:

The span of $\mathbf{y}$ and $\mathbf{z}$ defines a plane whose normal is parallel to $\mathbf{n} = \mathbf{y} \times \mathbf{z}$.

The equation of a plane is given by $\mathbf{x}\cdot\mathbf{n}=d$. If you're not familiar with this, the idea is that the component parallel to the normal, which is the dot product, is constant for all elements in the plane.

Multiplying out the dot product should give you the answer you need.