# 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

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

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

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

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

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 .

• -@User, you can use Mathpix Snip to convert mathematical terms in images to $\LaTeX$ code. – SarGe Aug 1 at 13:22

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 $$(u,v,w)\in V$$, $$\begin{vmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ u & v & w \\ \end{vmatrix} =0$$ $$\implies u(x_2y_3-y_2x_3)-v(x_1y_3-y_1x_3)+w(x_1y_2-x_2y_1) = 0 \implies \delta_{1} u-\delta_{2} v+\delta_{3} w=0$$
Hint: Note that the cross-product of $$x$$ and $$y$$ is given by $$x \times y = (\delta_1,-\delta_2,\delta_3).$$