How Do You Find the Intersection of a Line and a Plane?

Substitute the parametric coordinates of the line into the equation of the plane, solve for the line’s parameter, and substitute that value back into the line. This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 3.18: Intersections of lines and planes, and may be tested on Papers 1, 2, or 3.

A line in three dimensions is commonly written as the vector equation

$$\mathbf{r}=\mathbf{a}+\lambda\mathbf{b},$$

where $\mathbf{a}$ is the position vector of a point on the line, $\mathbf{b}$ is its direction vector, and $\lambda$ is a parameter. Convert this into parametric equations for $x$, $y$, and $z$, then substitute them into the plane equation.

For example, consider

$$\mathbf{r}=\begin{pmatrix}1\\2\\-1\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\3\end{pmatrix}$$

and the plane $x+2y-z=7$. The line gives

$$x=1+2\lambda,\qquad y=2-\lambda,\qquad z=-1+3\lambda.$$

Substituting into the plane:

$$(1+2\lambda)+2(2-\lambda)-(-1+3\lambda)=7.$$

Therefore,

$$6-3\lambda=7\quad\Rightarrow\quad\lambda=-\frac{1}{3}.$$

Substitute this value back into the line:

$$(x,y,z)=\left(\frac{1}{3},\frac{7}{3},-2\right).$$

Hence, the line intersects the plane at $\left(\frac{1}{3},\frac{7}{3},-2\right)$.

This method works because an intersection point must satisfy both equations simultaneously. Equivalently, if the plane is written $\mathbf{n}\cdot\mathbf{r}=d$, substitute the line to obtain $\mathbf{n}\cdot\mathbf{a}+\lambda(\mathbf{n}\cdot\mathbf{b})=d$. Therefore,

$$\lambda=\frac{d-\mathbf{n}\cdot\mathbf{a}}{\mathbf{n}\cdot\mathbf{b}},$$

provided $\mathbf{n}\cdot\mathbf{b}\neq 0$. This dot-product form is efficient when the plane’s normal vector is immediately visible.

A common misconception is that every line must intersect a plane. A line may instead be parallel to the plane or contained within it.

In an IB response, the command term find requires relevant working. Show the parametric substitution, solve for $\lambda$, state the complete coordinate, and verify it in the plane equation if time permits.