Write the equation of the plane through M_1(2,5,-3) and the…

Write the equation of the plane through $M_1(2,5,-3)$ and the intersection of $\vec r\cdot(3\vec i-\vec j-\vec k)=5$ and $\vec r\cdot(\vec i+2\vec k)=0$.

Step 1: Solution

The scalar forms are $3x-y-z-5=0$ and $x+2z=0$, so the pencil is

\[ (3+\lambda)x-y+(2\lambda-1)z-5=0 . \]

Through $M_1$: $2(3+\lambda)-5-3(2\lambda-1)-5=-1-4\lambda=0$, so $\lambda=-\tfrac14$.

\[11x-4y-6z-20=0\qquad\big(\vec r\cdot(11\vec i-4\vec j-6\vec k)=20\big)\]

Check:

• $M_1$: $22-20+18-20=0$;
• the points $(0,-5,0)$ and $(-2,-12,1)$ of the common line both satisfy the equation.