Line parallel to plane

Question

See if the line e is parallel with to the plane $α$. If not, find the intersecting point. $$\begin{align} α: & \quad \quad x-3y+z+1=0 \\ e: & \quad \{x+y-z=3, 2x-y-4z=3\} \end{align}$$

Answer 1

If a line is parallel to the plane, consider the consequences: any vectors that are orthogonal to the plane must be orthogonal to the line. This turns out to be the condition we need to check.

To find the line, work with the system of equations as to single out a variable. Added both equations we get $3x-5z=6$ or $x = 2 + (5z)/3$. Using the first equation we can write $$y=3+z-x = 3+z-2-\frac{5z}{3} = 1 - \frac{2z}{3}.$$ The line is now parameterized by $z$ as $$\begin{align} x & = 2 + \frac{5z}{3}, \\ y & = 1 - \frac{2z}{3}, \\ z & = z. \end{align}$$

To finish, you need to take two points in the line, translate that into a vector with initial and end point as chosen and check if it's orthogonal to the orthogonal vector to the plane, found using the coefficients of the given equation.