Write the equation of the plane through A(1,0,0) , B(0,1,0)…

Write the equation of the plane through $A(1,0,0)$, $B(0,1,0)$, $C(0,0,1)$.

Explanation of Method

The plane through three points on the coordinate axes, found by three methods.

Step 1: Intercept form

The intercepts are $a=b=c=1$, so $\tfrac x1+\tfrac y1+\tfrac z1=1$.

Step 2: Cross product

\[ \vec n=\overrightarrow{AB}\times\overrightarrow{AC}=\{-1,1,0\}\times\{-1,0,1\}=\{1,1,1\},\qquad 1(x-1)+y+z=0. \]

Step 3: Determinant

\[ \begin{vmatrix}x-1&y&z\\-1&1&0\\-1&0&1\end{vmatrix}=(x-1)+y+z=0 . \]

\[x+y+z=1\]

Check: $A$, $B$ and $C$ each give $1$. The normal $\{1,1,1\}$ makes equal angles $\arccos\tfrac1{\sqrt3}$ with the three axes. The distance from $O$ is $\tfrac1{\sqrt3}$.

$\vec n=\{1,1,1\}$ $A$ $B$ $C$

The plane $x+y+z=1$ cuts intercepts $1$ on all three axes; its normal $\{1,1,1\}$ makes equal angles with the axes.

Result

\[\boxed{\,x+y+z=1\,}\]