What are the intercepts of the planes $x = 0$ and $2y + 3z = 12$?

Question

What are the intercepts of the planes $x = 0$ and $2y + 3z = 12$? The word intercept is confusing me because I don't understand if I should say they intersect at point $(0,6,0)$ or the intercept is at $y=6$.

Answer 1

In R $^3$ , it is the line (x, y, z) = t(0,-3,2) + (0,6,0) You may find this by taking any vector (x, y, z) and asking when it satisfies both equations. The planes aren't parallel.

Answer 2

As @DavidPeterson as pointed out, the question alludes to the intersection of the two planes.

Now, you can use the fact that if the two planes intersect, then the intersection will be given by a straight line in space. Let the equation of the line be given by: $$ \mathbf{r}(t) = \mathbf{r}_0+t\mathbf{v}$$ where $\mathbf{r}_0$ is any point that lies on the line of intersection of the planes and $\mathbf{v}$ is the direction vector of the line. Also, let $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ be the cartesian unit vectors.

To find $\mathbf{v}$, use the fact that the direction of the line will be perpendicular to the normals of both planes.

For the plane $x=0$, one form of its direction vector (in the positive $x$-axis) is given by $\mathbf{n_1}=\mathbf{i}$.

For the plane $2y + 3z = 12$, its direction vector is given by $\mathbf{n_2}=2\mathbf{j}+3\mathbf{k}$.

Then, the direction vector of the line of intersection is given by: \begin{align} \mathbf{v} &= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 0 \\ 0 & 2 & 3 \\ \end{vmatrix} = -3\mathbf{j}+2\mathbf{k} \\ \end{align}

Therefore, the direction of the line of intersection is given by: \begin{align} \mathbf{v} &= \begin{pmatrix} 0 \\ -3 \\ 2 \\ \end{pmatrix} \nonumber \end{align}

Finally, a point on the line that satisfies both planes is $(x, y, z)=(0, 6, 0)$, or $(x, y, z) = (0, 0, 4)$.

Therefore, the equation of the line of intersection of the planes is: $$ \begin{align} \mathbf{r}(t) &= \begin{pmatrix} 0 \\ 6 \\ 0 \\ \end{pmatrix} + t\begin{pmatrix} 0 \\ -3 \\ 2 \\ \end{pmatrix} \end{align} \nonumber $$