Intersections and angles between lines and planes (HL only)

IB Mathematics AA HL· Geometry & Trigonometry: 3D Vectors· 15 min read

1. Intersection of a Line and a Plane★★★☆☆⏱ 5 min

📘 Definition

Three possible cases for line-plane intersection

Given a line $l$ with parametric form and plane $e x t Π$ with Cartesian equation, substitution gives a linear equation in the line parameter $t$ : 1. Unique solution for $t$ = one intersection point, 2. $0 t = k, k \neq = 0$ = line parallel to plane, no intersection, 3. $0 t = 0$ = line lies entirely on the plane, infinite intersections

📐 Worked Example

Find the intersection point of the line $r = 123 + t 2 - 1 1$ and plane $2 x + y - 2 z = 5$

1
Write parametric equations for the line from the vector form:
2
$x = 1 + 2t, \quad y = 2 - t, \quad z = 3 + t$
3
Substitute into the plane's Cartesian equation:
4
$2(1+2t) + (2-t) - 2(3 + t) = 5$
5
Expand and simplify to solve for $t$ :
6
$2 + 4t + 2 - t - 6 - 2t = 5 \implies -2 + t = 5 \implies t = 7$
7
Substitute $t = 7$ back into the parametric equations to get the intersection point:
8
$x = 1 + 14 = 15, \quad y = 2 - 7 = -5, \quad z = 3 + 7 = 10$
9
Verify the point satisfies the plane equation: $2 (15) + (- 5) - 2 (10) = 30 - 5 - 20 = 5$ , which matches, so the intersection is:
10
$(15, -5, 10)$

Exam tip:

Always check which case you have before concluding there is no intersection or an infinite number of intersections.

2. Angle Between a Line and a Plane★★★★☆⏱ 5 min

📘 Definition

Line-plane angle

The acute angle $θ$ (between $0^{\circ}$ and $9 0^{\circ}$ ) between a line and its orthogonal projection onto the plane. The formula uses the direction vector of the line $d$ and the normal vector of the plane $n$ .

The angle between the line and the plane's normal is $9 0^{\circ} - θ$ , so using the dot product formula for the angle between two vectors gives the identity $cos (9 0^{\circ} - θ) = sin θ = \frac{∣ d \cdot n ∣}{∣ d ∣∣ n ∣}$ . The absolute value ensures we get an acute angle.

📐 Worked Example

Find the acute angle between the line $r = 123 + t 2 - 1 1$ and plane $2 x + y - 2 z = 5$

1
Identify the direction vector of the line and normal vector of the plane:
2
$\vec{d} = \begin{pmatrix}2 \\ -1 \\ 1\end{pmatrix}, \quad \vec{n} = \begin{pmatrix}2 \\ 1 \\ -2\end{pmatrix}$
3
Calculate the dot product and magnitudes:
4
$\vec{d} \cdot \vec{n} = (2)(2) + (-1)(1) + (1)(-2) = 1, \quad |\vec{d}| = \sqrt{2^2 + (-1)^2 + 1^2} = \sqrt{6}, \quad |\vec{n}| = 3$
5
Substitute into the line-plane angle formula:
6
$\sin\theta = \frac{|1|}{\sqrt{6} \times 3} = \frac{1}{3\sqrt{6}} \approx 0.136$
7
Calculate $θ$ :
8
$\theta = \arcsin(0.136) \approx 7.8^\circ \quad (\text{or } 0.136 \text{ radians})$

3. Angle Between Two Intersecting Planes★★★★☆⏱ 5 min

📘 Definition

Dihedral angle (angle between two planes)

The acute angle between two intersecting planes, equal to the acute angle between their normal vectors. For planes with normals $n_{1}$ and $n_{2}$ , the formula is $cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣∣ n _{2} ∣}$ .

📐 Worked Example

Find the acute angle between planes $Π_{1} : x + y - z = 2$ and $Π_{2} : 2 x - y + 3 z = 1$

1
Extract the normal vectors from the Cartesian equations:
2
$\vec{n_1} = \begin{pmatrix}1 \\ 1 \\ -1\end{pmatrix}, \quad \vec{n_2} = \begin{pmatrix}2 \\ -1 \\ 3\end{pmatrix}$
3
Calculate dot product and magnitudes:
4
$\vec{n_1} \cdot \vec{n_2} = (1)(2) + (1)(-1) + (-1)(3) = -2, \quad |\vec{n_1}| = \sqrt{3}, \quad |\vec{n_2}| = \sqrt{14}$
5
Substitute into the formula (remember absolute value):
6
$\cos\theta = \frac{| -2 |}{\sqrt{3} \sqrt{14}} = \frac{2}{\sqrt{42}} \approx 0.309$
7
Calculate $θ$ :
8
$\theta = \arccos(0.309) \approx 72^\circ \quad (\text{or } 1.26 \text{ radians})$

IB exam questions almost always ask for the acute angle between two planes. Only omit the absolute value if explicitly asked for the obtuse angle.

4. Common Pitfalls

Wrong move:

Using $cos θ$ instead of $sin θ$ for the line-plane angle

Why:

You confuse the angle between the line and the normal with the line-plane angle itself

Correct move:

Memorize that line-plane angle uses $sin θ = \frac{∣ d \cdot n ∣}{∣ d ∣∣ n ∣}$

Wrong move:

Forgetting the absolute value around the dot product, resulting in an obtuse angle when acute is required

Why:

The dot product of normals can be negative, which means the angle between the normals is obtuse; the dihedral angle is always acute unless stated otherwise

Correct move:

Always add an absolute value around the dot product in all angle formulas for this topic

Wrong move:

Stopping after solving for $t$ and not reporting the intersection point coordinates

Why:

Rushing in the exam, you misread the question which asks for the intersection, not just the parameter

Correct move:

Always substitute the calculated $t$ back into the parametric equation of the line to get the final intersection coordinates

Wrong move:

Calling any inconsistent result 'no intersection' without checking if the line is on the plane

Why:

You forget the three possible cases for line-plane intersection

Correct move:

If you get $0 t = k$ : if $k \neq = 0$ , it is parallel with no intersection; if $k = 0$ , the line lies on the plane with infinite intersections

5. Quick Reference Cheatsheet

Concept	Formula/Method	Key Notes
Line-plane intersection	Substitute $x (t), y (t), z (t)$ into plane equation	3 cases: 1 solution = 1 point, $0 = k \neq = 0$ = parallel, $0 = 0$ = line on plane
Angle: line-plane	$sin θ = \frac{∣ d \cdot n ∣}{∣ d ∣∣ n ∣}$	$d$ = line direction, $n$ = plane normal, acute only
Angle: two planes	$cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣∣ n _{2} ∣}$	Dihedral angle, always acute unless stated otherwise

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.

2022 · Paper 1
Find angle between line and plane
2023 · Paper 2
Find intersection and dihedral angle

Going deeper

syllabusIB AA HL 3D Geometry SyllabusOfficial HL topic reference

What's Next

Mastering intersections and angles between lines and planes is a critical foundation for all advanced 3D vector geometry topics that appear frequently on IB AA HL exams. These skills are used to solve problems involving shortest distances between points, lines and planes, and to find the intersection of multiple planes, which are common long questions in both paper 1 and paper 2. They also underpin more advanced HL topics including vector integration in the calculus option, and 3D shape volume problems. Building fluency with these formulas and methods will save you significant time in the exam.