Find vector and parametric equations of the line passing through the given point and perpendicular to the given plane.

Question

Analytical Geometry of Space – Equation of Line

Question: Find vector equation, parametric equations of the line passing through the point $(5,1,0)$ and perpendicular to the plane $2x-y+z=1$. Also find two other points on the line.

Answer 1

Equation of a Line Through a Point and Perpendicular to a Plane

The equation of a line passing through the point $\, P_1=(x_1,y_1,z_1)\,$ perpendicular to a plane given by the equation $\, Ax+By+Cz+D=0\,$ or by $\, \vec{r}\vec{N}+D=0\,$ is
$\dfrac{x-x_1}{A}=\dfrac{y-y_1}{B}=\dfrac{z-z_1}{C}\,$ (in component form), and

$(\vec{r} - \vec{r_1}) \times \vec{N} = \vec{0}\, $ in vector form.

Once you have a line, it is trivial to find any other two points on it...