3D planes, intersections of lines and planes

IB Mathematics: Applications and Interpretation HL· 35 min read

1. Equations of a 3D Plane★★☆☆☆⏱ 10 min

A plane is a flat, 2-dimensional surface that extends infinitely in 3D space. It can be written in two commonly used forms for IB exams: Cartesian form and vector form, both defined using a point on the plane and a normal vector.

📘 Definition

Normal Vector to a Plane

$n = a b c$

A non-zero vector that is perpendicular to every direction vector that lies on the plane

Example:

A plane that contains the entire x-axis has normal vector $010$

\text{Cartesian form: } ax + by + cz = d, \quad d = \vec{n} \cdot \vec{r}_0

\text{Vector form: } \vec{r} = \vec{r}_0 + \lambda \vec{v}_1 + \mu \vec{v}_2

📐 Worked Example

Find the Cartesian equation of the plane that passes through $(2, 1, 0)$ and has normal vector $13 - 2$

1
Start with the general Cartesian form, substitute values for $a, b, c$ from the normal vector:
2
$1x + 3y - 2z = d$
3
Substitute the point $(2, 1, 0)$ to calculate $d$ :
4
$d = (1)(2) + (3)(1) + (-2)(0) = 5$
5
Write the final equation:
6
$x + 3y - 2z = 5$

2. Unique Intersection of a Line and Plane★★★☆☆⏱ 15 min

When a line is not parallel to a plane, it will cross the plane at exactly one point. The method to find this point is straightforward: substitute the parametric equations of the line into the Cartesian equation of the plane, solve for the parameter, then substitute back to find coordinates.

📐 Worked Example

Find the intersection point of the line $x = 1 + λ$ , $y = - λ$ , $z = 2 + 2 λ$ and the plane $x + 2 y - z = 5$ .

1
Substitute the parametric expressions for $x, y, z$ into the plane equation:
2
$(1 + \lambda) + 2(-\lambda) - (2 + 2\lambda) = 5$
3
Simplify and solve for $λ$ :
4
$1 + \lambda - 2\lambda - 2 - 2\lambda = 5 \implies -3\lambda - 1 = 5 \implies \lambda = -2$
5
Substitute $λ = - 2$ back to get the coordinates:
6
$x = 1 - 2 = -1, \quad y = -(-2) = 2, \quad z = 2 + 2(-2) = -2$
7
Final intersection point: $(- 1, 2, - 2)$

3. Special Cases: No or Infinite Intersections★★★★☆⏱ 15 min

When the direction vector of the line is perpendicular to the plane's normal vector, the line is parallel to the plane. This gives two possible outcomes:

No intersection: The line is parallel but does not lie on the plane. There is no solution.
Infinite intersections: The entire line lies on the plane. Every point on the line is an intersection.

📐 Worked Example

Classify the intersection of the line $r = 102 + λ 111$ and the plane $2 x - 4 y + 2 z = 10$ .

1
Identify the direction vector of the line $d = 111$ and the normal vector of the plane $n = 2 - 4 2$ .
2
Calculate the dot product to check for parallelism:
3
$\vec{d} \cdot \vec{n} = (1)(2) + (1)(-4) + (1)(2) = 0$
4
Dot product is zero, so the line is parallel. Check if the known point on the line lies on the plane:
5
$2(1) - 4(0) + 2(2) = 2 + 0 + 4 = 6 \neq 10$
6
Conclusion: The line is parallel to the plane, but no point of the line lies on the plane, so there are no intersections.

4. Contextual Applications★★★★☆⏱ 10 min

Line-plane intersections are common in real-world contexts tested in IB AI HL, including projectile motion, drone navigation, and 3D manufacturing design. The method for solving these problems follows the same steps as abstract geometry problems.

📐 Worked Example

A drone flies along a path with position at time $t$ seconds given by $(t, 2 t, 10 - t)$ (coordinates in meters). The landing pad lies on the plane $z = 0$ . Find the time when the drone lands and its landing position.

1
The intersection with the landing pad occurs when $z = 0$ , so set the z-coordinate equal to 0:
2
$10 - t = 0 \implies t = 10 \text{ seconds}$
3
Substitute $t = 10$ back to find the landing coordinates:
4
$x = 10, \quad y = 2(10) = 20, \quad z = 0$
5
The drone lands after 10 seconds at position $(10, 20, 0)$ meters.

5. Common Pitfalls

Wrong move:

Assuming parallel lines always have no intersection

Why:

If the entire line lies on the plane, there are infinite intersections, a common exam trick question

Correct move:

After confirming $d \cdot n = 0$ , test one point on the line to check if it lies on the plane

Wrong move:

Sign errors when calculating $d$ for the plane equation

Why:

Mistakes when expanding $a (x - x_{0}) + b (y - y_{0}) + c (z - z_{0}) = 0$ often flip the sign of $d$

Correct move:

Always substitute the original point back into the final equation to check $d$ after calculation

Wrong move:

Mixing up normal and direction vectors

Why:

Confusing which vector is perpendicular to the plane leads to wrong dot product results

Correct move:

Remember: Normal = perpendicular to the plane, direction vectors = parallel to the plane

Wrong move:

Stopping after finding the parameter, not checking for consistency

Why:

You can get an incorrect value for the parameter if you forget to check parallel cases, leading to a false intersection point

Correct move:

Always check the dot product first, or verify that your final point satisfies the plane equation

6. Quick Reference Cheatsheet

Intersection Case	Vector Condition	Number of Solutions
Line crosses plane	$d \cdot n \neq = 0$	1 unique point
Line parallel, not on plane	$d \cdot n = 0$ , point $\in /$ plane	0 intersections
Line lies on plane	$d \cdot n = 0$ , point $\in$ plane	Infinite intersections

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 2
Find intersection of line and plane
2021 · 1
Classify line-plane intersection case

What's Next

Mastering line-plane intersections builds the foundation for more complex 3D geometry topics in IB AI HL, including intersections of multiple planes, which relies on the same vector logic and understanding of solution cases for linear systems. This topic is also frequently combined with vector application problems like kinematics in 3D space, so getting comfortable with classifying cases will save you time on multi-part exam questions.