I would like to ask about 3R Planes. [Hi, I have a question in my homework and I attached my solution. By any chance somebody check is it a good solution and correct answer? Thank you.][1]
[1]: https://i.stack.imgur.com/XcHWC.jpg![enter image description here](https://i.stack.imgur.com/X9fgl.jpg)
Your solutions are correct. A tip on how to check these answers for yourself: dot your solution line vector with the normals of the planes. If you get zero, it means they are perpendicular, so the line is in the plane (if at least one point on the line is in the plane).
Details: Your solution has vector $(-1,1,1)$, while your plane $\pi_1$ has normal vector $(1,2,-1)$. And note that $(-2,1,0)$ is on your line, but obviously that satisfies $x+2y-1z=0$.
Your solutions are correct. A tip on how to check these answers for yourself: dot your solution line vector with the normals of the planes. If you get zero, it means they are perpendicular, so the line is in the plane (if at least one point on the line is in the plane).
Details: Your solution has vector $(-1,1,1)$, while your plane $\pi_1$ has normal vector $(1,2,-1)$. And note that $(-2,1,0)$ is on your line, but obviously that satisfies $x+2y-1z=0$.