Find what is span of 2 linearly independent vectors is I have been trying assignment questions of linear algebra and I am unable to solve this particular question

Let $x=\left(x_{1}, x_{2}, x_{3}\right), y=\left(y_{1}, y_{2}, y_{3}\right) \in \mathbb{R}^{3}$
be linearly independent. Let $\delta_{1}=x_{2} y_{3}-y_{2} x_{3}, \delta_{2}=x_{1} y_{3}-y_{1} x_{3}$
$\delta_{3}=x_{1} y_{2}-y_{1} x_{2} .$ If $V$ is the span of $x, y$ then

*

*$V=\left\{(u, v, w): \delta_{1} u-\delta_{2} v+\delta_{3} w=0\right\}$


*$V=\left\{(u, v, w):-\delta_{1} u+\delta_{2} v+\delta_{3} w=0\right\}$


*$V=\left\{(u, v, w): \delta_{1} u+\delta_{2} v-\delta_{3} w=0\right\}$


*$V=\left\{(u, v, w): \delta_{1} u+\delta_{2} v+\delta_{3} w=0\right\}$

I know the definitions of Linearly Independent and Span of vectors but I don't know how to solve this problem due to $\delta_{1}$, $\delta_{2}$,$\delta_{3}$ as I am unable to write V in terms of $\delta_{i}$'s and $(u, v, w)$.
Any help will be really appreciated .

Given, $x=(x_1,x_2,x_3), y=(y_1,y_2,y_3) \in \mathbb{R}^3 $ are linearly independent.
Also, given, $V$ is the span of that two linearly independent vectors $x,y\in \mathbb{R}^3 $
It means clearly that each $(u,v,w)\in V $ can be uniquely spanned by the linearly independent vectors $x,y\in \mathbb{R}^3 $
That means for each $(u,v,w)\in V $,
$$
    \begin{vmatrix}
    x_1 & x_2 & x_3 \\
    y_1 & y_2 & y_3 \\
    u & v & w \\
    \end{vmatrix}
    =0 
$$
$\implies u(x_2y_3-y_2x_3)-v(x_1y_3-y_1x_3)+w(x_1y_2-x_2y_1) = 0 \implies \delta_{1} u-\delta_{2} v+\delta_{3} w=0 $
