Suppose $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ lie on a plane through the origin.

Question

What determinant is zero? What equation does this give for the plane?

I need some help here, am pretty stuck

Answer 1

Hint: the matrix with rows $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ would have determinant $0$, since the three position vectors, from the origin to each of these three points, are coplanar.

Answer 2

If you have a matrix $M$ composed of vectors $v_1, v_2,... v_n$:

$$M = \begin{bmatrix} \uparrow & \uparrow & ... & \uparrow \\ v_1 & v_2 & ... & v_n\\ \downarrow & \downarrow & ... & \downarrow\end{bmatrix}$$

Then one consequence of $M$ having a determinant of zero is that the vectors $v_1, v_2,... v_n$ are not linearly independent. That is, there are some real numbers $C_1 ... C_n$ such that $C_1v_1 + C_2v_2 + ... + C_nv_n = 0$.

Next...

Suppose you have 2 points, $A$ and $B$, both in a plane. What can you say about point $A+B$?

If you need more help than this let me know in a comment. If this isn't homework (such as self study) and you want a complete solution instead of a hint then also comment that.