Find the parametric equation for a line through a point and parallel to two planes

by RoRi

May 30, 2016

We say that a line is parallel to a plane if the direction vector of the line is parallel to the plane. Let $L$ be the line containing the point $(1,2,3)$ and parallel to the planes

$x + 2y + 3z = 4, \qquad 2x + 3y + 4z = 5.$

Find a vector parametric equation for $L$ .

The normal vectors of the planes are $N_1 = (1,2,3)$ and $N_2 = (2,3,4)$ . So, the direction vector $A = (a_1, a_2, a_3)$ of $L$ will be perpendicular to both of these,

$\begin{align*} N_1 \cdot A &= 0 & \implies && a_1 + 2a_2 + 3a_3 &= 0 \\ N_2 \cdot A &= 0 & \implies && 2a_1 + 3a_2 + 4a_3 &= 0. \end{align*}$

From the first equation we have $x = -2y - 3z$ . Plugging this into the second equation we obtain $y = -2z$ , which then gives us $x = z$ . Since $z$ is arbitrary, we take $z = 1$ to obtain a direction vector $A = (1,-2,1)$ . Therefore, the vector parametric equation for the line is