Consider the plane $P_1 : 3x − 5y + 2z = 1$ in $\Bbb{R^3}$

Question

a) Give a vector equation of $P_1$.

b) Find the point of intersection of $P_1$ and the line $l$ whose parametric equations are $x = 5 + t , \ y = 2 − 2t , \ z = −1 − 6t$.

c) Find a general equation of the plane $P_2$ in $\Bbb{R^3}$ which passes through the point $p = (1, −2, 3)$ and is parallel to $P_1$.

I'm very uncertain of how to tackle these questions, and have been unable to find anything on them. I would greatly appreciate it if someone would explain how to go about solving these sorts of questions.

Answer 1

HINT

• for point $b)$ plug in $P_1$ $x$ $y$ and $z$ to find $t$
• for point $c)$ recall that the parallel plane has the same normal vector that is

$$3x − 5y + 2z = d$$