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The Intersection Between Planes and Coordinate Axes

Along an axis, the other coordinates always equal $0$ . For that reason, the rule below is important to remember when you are trying to find where planes intersect the different axes.

Rule

Intersections with Coordinate Axes

If you want to find the point of intersection with the

$x$ -axis, you set $y = 0$ and $z = 0$ in the equation of the plane and solve for $x$ .
$y$ -axis, you set $x = 0$ and $z = 0$ in the equation of the plane and solve for $y$ .
$z$ -axis, you set $x = 0$ and $y = 0$ in the equation of the plane and solve for $z$ .

Example 1

Given the plane $x - 3 y + 2 z + 2 = 0$ , find the intersections with the coordinate axes.

The intersection with the $x$ -axis: Set $y = 0$ and $z = 0$ in the equation of the plane and solve for $x$ : $\begin{array}{l} x - 3 \cdot 0 + 2 \cdot 0 + 2 & = 0, \\ x + 2 & = 0, \\ x & = - 2 . \end{array}$

This means the plane intersects the $x$ -axis at the point $(- 2, 0, 0)$ . Intersection with the $y$ -axis: Sett $x = 0$ and $z = 0$ in the equation of the plane and solve for $y$ : $\begin{array}{l} 0 - 3 y + 2 \cdot 0 + 2 & = 0, \\ - 3 y + 2 & = 0, \\ y = \frac{2}{3} . \end{array}$

That means the intersection with the $y$ -axis is at the point $(0, \frac{2}{3}, 0)$ . Intersection with the $z$ -axis: Set $x = 0$ and $y = 0$ in the equation of the plane and solve for $z$ : $\begin{array}{l} 0 - 3 \cdot 0 + 2 z + 2 & = 0, \\ 2 z + 2 & = 0, \\ z & = - 1 . \end{array}$

This means the plane intersects the $z$ -axis at the point $(0, 0, - 1)$ . Sometimes, one of these equations have no solution. In that case, the plane is parallel to one of the axes. You can confirm this by looking at the normal vector to the plane. The normal vector will be $0$ along the axes that are parallel to the plane.

Example 2

Given the plane $x - 3 = 0$ , find the intersection with the coordinate axes.

Here, you can see that the normal vector is $(1, 0, 0)$ . That means this plane will be parallel to both the $y$ -axis and the $z$ -axis. To find the intersection with the $x$ -axis, you set $y = 0$ and $z = 0$ and solve the equation for the plane. That gives you $x - 3 = 0$ , which means that $x = 3$ , and the intersection is at the point $(3, 0, 0)$ .

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