True or false? The three points $(1,2,0), (1,1,1)$ and $(0,0,-1)$ lie in the same plane in $\mathbb{R}^{3}$
I think an equivalent question would be if these vectors are linearly dependent?
So I made this in matrix form and used sarrus rule to calculate the determinant.
$$\begin{vmatrix} 1 & 1 & 0\\ 2 & 1 & 0\\ 0 & 1 & -1 \end{vmatrix}\left.\begin{matrix} 1 & 1 & \\ 2 & 1 & \\ 0 & 1 & \end{matrix}\right|$$
And the determinant is $-1$ and thus the statement is false.
Did I do it correctly? Please tell me, I'm preparing for an exam.
The thing you did wrong is that, considering these points as vectors you add the starting point $(0,0,0)$, so you try to see that these points and the point $(0,0,0)$ is on the same plane or not. And your result shows that they are not. You may choose one point as a starting point, then you have 2 vectors, which must lie in a plane. In general, if you have more than 3 points, you may choose one of them, say A, as a starting point, then calculate the vectors starting with A and ending with each of other points, then see if these vectors belong to the same plane or not.