Exercise 30

Find the plane that goes through the points $(1,2,3)$, $(4,5,6)$, and $(7,8,9)$ or explain why there is not one plane that satisfies this condition.

Find all planes in $\mathbb{R}^3$ whose line of intersection with the $xz$-plane is given by $\pi x +e z=\sqrt{5}$.

Give two vectors that are in the plane given by $3x+4y=6z-20$ and are orthogonal to each other.

Give the equations of two planes that make a $90^\circ$ angle with $3x+4y=6z-20$ and have an angle of $90^\circ$ between them as well.

Give the parametric form of the line of intersection for the planes given by $x+y+z=10$ and $-2x-4y+2z=7$.