Write the equation of the plane through M_1(2,-1,1)…

Write the equation of the plane through $M_1(2,-1,1)$ perpendicular to both $3x+2y-z+4=0$ and $x+y+z-3=0$.

Explanation of Method

The plane through M₁(2, −1, 1) perpendicular to two given planes.

Step 1: Solution

The normal $\vec n$ must be perpendicular to both $\vec n_1=\{3,2,-1\}$ and $\vec n_2=\{1,1,1\}$:

\[ \vec n=\vec n_1\times\vec n_2=\{2+1,\ -(3+1),\ 3-2\}=\{3,-4,1\}. \]

\[3x-4y+z-11=0\]

Check: $M_1$ gives $6+4+1-11=0$; $\vec n\cdot\vec n_1=9-8-1=0$ and $\vec n\cdot\vec n_2=3-4+1=0$.

Result

\[\boxed{\,3x-4y+z-11=0\,}\]

Remark

Wording correction: the source says $\vec n$ “must be parallel to both $\vec n_1$ and $\vec n_2$”. It must be perpendicular to both, which is why it is their cross product. The required plane is also perpendicular to the line $\pi_1\cap\pi_2$.