To solve this problem, we need to determine which points lie in the plane defined by three given points $A(2, 0, 3)$, $B(4, 0, 1)$, and $C(1, -2, 0)$.

Step 1: Find the equation of the plane

The general form of a plane equation is: $Ax + By + Cz + D = 0$ We can find the coefficients $A$, $B$, $C$, and $D$ using points $A$, $B$, and $C$.

Step 2: Find two vectors in the plane

Define two vectors:

• $\overrightarrow{AB} = B - A = (4 - 2, 0 - 0, 1 - 3) = (2, 0, -2)$
• $\overrightarrow{AC} = C - A = (1 - 2, -2 - 0, 0 - 3) = (-1, -2, -3)$

Step 3: Find the normal vector to the plane

The normal vector $\vec{n}$ to the plane is the cross product of $\overrightarrow{AB}$ and $\overrightarrow{AC}$: $\vec{n} = \overrightarrow{AB} \times \overrightarrow{AC}$ The cross product is computed as: