AHL 3.18—Intersections of lines & planes

In three-dimensional analytic geometry, understanding how lines and planes interact is essential for solving spatial problems. This subtopic covers the intersections of lines with planes, planes with planes, and three planes together , as well as the angles between them and distances between geometric objects.

Before diving in, recall the two key objects we work with:

• A line in 3D is typically expressed in parametric form: $\mathbf{r}=\mathbf{a}+t\mathbf{v}$, where $\mathbf{a}$ is a point on the line, $\mathbf{v}$ is the direction vector, and $t\in \mathbb{R}$.
• A plane in 3D can be expressed in two equivalent forms:
  • Cartesian form: $ax+by+cz=d$, where $\mathbf{n}=(a,b,c)$ is the normal vector.
  • Vector form: $\mathbf{r}\cdot \mathbf{n}=\mathbf{a}\cdot \mathbf{n}$, where $\mathbf{a}$ is any point on the plane. Both forms appear in IB exam questions.

Normal Vector: A vector $\mathbf{n}$ that is perpendicular to every vector lying within a given plane. It is used to define the orientation of the plane in 3D space.

Direction Vector: A vector $\mathbf{v}$ that indicates the direction in which a line travels. It is the key component in the parametric equation of a line.

When a line meets a plane in 3D space, exactly three outcomes are possible:

1. One point of intersection , the line passes through the plane at a unique point (the most common case). This occurs when $\mathbf{v}\cdot \mathbf{n}\ne 0$.
2. No intersection , the line is parallel to the plane: $\mathbf{v}\cdot \mathbf{n}=0$, and the point $\mathbf{a}$ on the line does not satisfy the plane equation.
3. Infinitely many intersections , the line lies entirely within the plane: $\mathbf{v}\cdot \mathbf{n}=0$ and every point on the line satisfies the plane equation.

Distinguishing cases 2 and 3: If $\mathbf{v}\cdot \mathbf{n}=0$, substitute any point on the line into the plane equation. If it satisfies the equation, the line lies in the plane (case 3); if not, the line is parallel but not in the plane (case 2).

Method: Substitution

To find the intersection point, substitute the parametric equations of the line into the Cartesian equation of the plane and solve for the parameter $t$. Then substitute $t$ back into the line equations to find the coordinates.