Let $x = [3, 4, 2], y = [2, −1, 3],$ and $z = [−1, 2, 1]$.
Give a general equation of the plane $P$ in $\mathbb{R}^3$ which passes through the point $[1, −2, 2]$ and has direction vectors $x$ and $y$.
I started off by putting them in an augmented matrix $[A|B]$ but I later found out it is inconsistent. Am I supposed to use cross products for this question? Any help appreciated!
When given three points, to find the equation of a plane, we need to find two vectors in the plane, take their cross product to get a normal vector to the plane. I am not sure why the $z$ vector is given, as we only need the $x$ and $y$ vectors and the other specific vector that we need the plane to pass through.
Let $\vec{X}=[x_1,y_1,z_1], \vec{Y}=[x_2,y_2,z_2]$, and $\vec{P}=[x_0,y_0,z_0]$. These are analogous to the $x$, $y$, and the specific point, $p$, that were given in your question. Now we need two points in the plane, which we can obtain by subtracting $\vec{Y}$ from $\vec{X}$ and $\vec{P}$ from $\vec{X}$.
$\vec{U}=\vec{X}-\vec{Y}=[x_1-x_2,y_1-y_2,z_1-z_2]$ and $\vec{V}=\vec{X}-\vec{P}=[x_1-x_0,y_1-y_0,z_1-z_0]$.
The cross product of these two vectors will give a normal vector to the plane.
$$\vec{N}=\vec{U}\times\vec{V}=\det\begin{vmatrix}i&j&k\\x_1-x_2&y_1-y_2&z_1-z_2\\x_1-x_0&y_1-y_0&z_1-z_0 \end{vmatrix}=Ai+Bj+Ck $$ By plugging in specific numbers or actually expanding (although I do not recommend, as the symbols can become confusing quickly), then you can figure out what the values of $A,B$, and $C$ are.
Note that any vector whose dot product with $\vec{N}$ is $0$ is perpendicular to $\vec{N}$. So, $\vec{N}\bullet(\vec{R}-\vec{P})=0$ where $\vec{R}=[x,y,z]$ -- points in the plane, which make the dot product with $\vec{N}=0$ -- will suffice because there is a whole plane of vectors orthogonal to $\vec{N}$. $$\vec{N}\bullet(\vec{R}-\vec{P})=A(x-x_0)+B(y-y_0)+C(z-z_0)=0 $$
Plugging your points into this will give you the formula for your plane. I hope this makes sense. It really helps if you draw out what is being described.
If this didn't help, here are some links that I found helpful when learning this: Paul's Math Notes Wikipedia Brilliant