Find parametric equations of a plane

Question

a) Find parametric equations of the line that goes through $ A (3,6,4) $, intersects the $ Oz $ axis, and is parallel to the plane $\pi: \ x-3y+5z-6=0$;

b) The plane passing $ A (-1,2,5) $ and is perpendicular to the intersection of the $ \pi_1:\ 2x-y + 3z-4 = 0 $ and $ \pi_2: \ x + 2y-4z + 1 = 0$

How can I solve these types of problems?

Answer 1

HINT

For the point "a" we can

• consider a direction vector $\vec v$ for the line that goes through $A (3,6,4)$ and $P(0,0,t)$
• the line is parallel to the plane for $\vec n\cdot \vec v=0$ where $\vec n$ is a normal vector to the plane

For the point "b" we have that

• a direction vector $\vec v$ for the line intersection of $\pi_1$ and $\pi_2$ is $\vec v=\vec n_1\times \vec n_2$
• $\vec v$ is also a normal vector for the plane perpendicular to the intersection of $\pi_1$ and $\pi_2$