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IB DP Maths AA · Topic 3 Geometry and trigonometry · AHL 3.12–3.18 · ~10 min read

Vectors, lines and planes

HL vectors in full: column and i, j, k notation, magnitude and unit vectors, the scalar product and angles, vector equations of lines, intersecting, parallel and skew lines, the vector product, equations of planes, and intersections and angles between lines and planes.

Vector notationMagnitude & unit vectorsPosition vectorsScalar (dot) productVector equation of a lineSkew linesVector (cross) productEquations of a plane
RSby Riyaz Sir
mentorriyaz.in
No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • A vector has size and direction. AB = b − a (end minus start), and |v| = √(x2 + y2 + z2).
  • Scalar (dot) product: v · w = v1w1 + v2w2 + v3w3 = |v||w| cos θ. It is 0 exactly when the vectors are perpendicular.
  • Line: r = a + λb (point a, direction b). In 3D two lines are parallel, intersecting or skew.
  • Vector (cross) product: v × w is perpendicular to both v and w, and |v × w| = |v||w| sin θ is the area of the parallelogram.
  • Plane: r · n = a · n, or ax + by + cz = d, where n = (a, b, c) is the normal vector.
  • This whole note is HL only (AHL 3.12–3.18) and unchanged for 2029.

By the end you can

  • Use vector notation, find magnitudes and unit vectors, and work with position vectors. HL
  • Use the scalar product to find angles and test for perpendicular vectors. HL
  • Write vector, parametric and Cartesian equations of lines, and decide whether two lines are parallel, intersecting or skew. HL
  • Find and use the vector product, including for areas. HL
  • Write equations of planes in vector and Cartesian form. HL
  • Find intersections of lines and planes and of two or three planes, and the angles between them. HL

1Vector basics

xyzP(3, 2, 4)OP = 3i + 2j + 4k|OP| = √(9 + 4 + 16)= √29AB = b − a(end − start)
A position vector goes from the origin O to a point; AB goes from A to B.
|v| = √(v12 + v22 + v32) FORMULA BOOKLET
magnitude
  • Unit vector in the direction of v: v/|v|. Parallel vectors are multiples of each other: w = kv.
  • Notation: (3, 2, 4) as a column vector is the same as 3i + 2j + 4k. Write vectors with an underline or arrow in your working.
Worked example

A(1, 2, −1) and B(4, −2, 11). Find AB and a unit vector in the direction of AB.

  1. AB = b − a = (3, −4, 12) A1
  2. |AB| = √(9 + 16 + 144) = 13 A1
  3. unit vector 113(3, −4, 12) A1
  • Dividing a line: the point P that divides AB in the ratio 1 : 2 has p = a + ⅓AB.

2The scalar product

v · w = v1w1 + v2w2 + v3w3, cos θ = v · wǀvǀǀwǀ FORMULA BOOKLET
angle between two vectors
Worked example

Find the angle between a = (2, −1, 3) and b = (1, 4, −2).

  1. a · b = 2 − 4 − 6 = −8 A1
  2. cos θ = −8√14 √21 M1
  3. θ = 118° A1
Worked example

Find p if (p, 2, −1) and (3, p, 4) are perpendicular.

  1. 3p + 2p − 4 = 0 M1
  2. p = 0.8 A1
Exam tipNegative dot product means the angle is obtuse. For the angle between two lines, take the acute angle: use |v · w|.

3Equations of lines

r = a + λb FORMULA BOOKLET
a = a point on the line, b = direction vector
  • Parametric form: x = a1 + λb1, y = a2 + λb2, z = a3 + λb3. Cartesian form: x − a1b1 = y − a2b2 = z − a3b3.
  • Kinematics: r = r0 + tv describes a particle moving with constant velocity v; speed = |v|. Two particles collide only if they are at the same point at the same time.
intersectingskew: not parallel, never meetLines in 3D: parallel, intersecting, or skew
In 2D, lines that are not parallel must meet. In 3D they can miss each other completely.
  • The test: if the directions are multiples, the lines are parallel (or the same line). If not, equate the components: if one λ and μ satisfy all three equations they intersect; if not, they are skew.
Worked example

L1: r = (1, 2, 3) + λ(2, −1, 1) and L2: r = (2, −3, 5) + μ(1, 1, 0). Show that they intersect and find the point.

  1. x: 1 + 2λ = 2 + μ; y: 2 − λ = −3 + μ M1
  2. subtract: 3λ − 1 = 5 ⇒ λ = 2, μ = 3 A1
  3. z: 3 + 2 = 5 ✓, so they meet at (5, 0, 5) R1A1
Worked example

Find the acute angle between L1 and L2.

  1. cos θ = ǀ2 − 1 + 0ǀ√6 √2 = 1√12 M1
  2. θ = 73.2° A1
Common mistakeChecking only two of the three equations. Two equations always have a solution for λ and μ; the third one decides whether the lines meet.

4The vector product

aba × bθa × b ⟂ both a and b|a × b| = |a||b| sin θ= area of theparallelogram
Use the right-hand rule: a along your first finger, b along your second, a × b along your thumb.
v × w = (v2w3 − v3w2, v3w1 − v1w3, v1w2 − v2w1) FORMULA BOOKLET
perpendicular to v and w
ǀv × wǀ = ǀvǀǀwǀ sin θ FORMULA BOOKLET
area of parallelogram; half of it for a triangle
Worked example

Find (2, −1, 3) × (1, 4, −2).

  1. x: (−1)(−2) − (3)(4) = −10 M1
  2. y: (3)(1) − (2)(−2) = 7; z: (2)(4) − (−1)(1) = 9 A1
  3. (−10, 7, 9) A1
  • Check: dot your answer with both original vectors. (−10, 7, 9) · (2, −1, 3) = −20 − 7 + 27 = 0 ✓.
  • Order matters: w × v = −(v × w).

5Equations of planes

A (point on plane)n (normal)r − an ⟂ (r − a) for every point R in the plane, so r · n = a · nCartesian form: ax + by + cz = d, where n = (a, b, c)
Any vector lying in the plane is perpendicular to the normal n.
r = a + λb + μc FORMULA BOOKLET
vector form: a point and two directions in the plane
r · n = a · n, ax + by + cz = d FORMULA BOOKLET
normal form and Cartesian form
Worked example

Find the Cartesian equation of the plane through A(1, 0, 2), B(2, 1, 0) and C(0, 3, 1).

  1. AB = (1, 1, −2), AC = (−1, 3, −1) A1
  2. n = AB × AC = (5, 3, 4) M1A1
  3. d = n · a = 5 + 0 + 8 = 13, so 5x + 3y + 4z = 13 A1
Worked example

Find the area of triangle ABC.

  1. area = ½|AB × AC| = ½√(25 + 9 + 16) M1
  2. = 5√22 ≈ 3.54 A1

6Intersections and angles

  • Line meets plane: substitute the parametric form of the line into the plane's equation and solve for λ.
Worked example

Find where r = (1, 1, 1) + λ(1, 0, 2) meets the plane x + 2y − z = 6.

  1. (1 + λ) + 2 − (1 + 2λ) = 6 M1
  2. λ = −4 A1
  3. (−3, 1, −7) A1
Worked example

Find the acute angle between that line and the plane.

  1. the angle with the normal: d · n = 1 + 0 − 2 = −1 M1
  2. sin θ = ǀ−1ǀ√5 √6 = 1√30 M1
  3. θ = 10.5° A1
Common mistakeUsing cos instead of sin for a line and a plane. The dot product gives the angle with the normal, which is 90° minus the angle with the plane, so use sin θ.
  • Two planes: their line of intersection has direction n1 × n2. Find one point on it by setting one coordinate (for example z = 0) and solving the other two equations.
Worked example

Find the line of intersection of x + y + z = 6 and x − y + 2z = 5.

  1. direction = (1, 1, 1) × (1, −1, 2) = (3, −1, −2) M1A1
  2. z = 0: x + y = 6, x − y = 5 ⇒ (5.5, 0.5, 0) A1
  3. r = (5.5, 0.5, 0) + λ(3, −1, −2) A1
  • Angle between two planes = the angle between their normals (take the acute one). Three planes can meet at a point, along a line (a sheaf), or not at all; solve the 3 × 3 system with the GDC and interpret.
GDC (calculator) methodPaper 2: the GDC solves 3 × 3 linear systems instantly. A "no solution" or "infinitely many solutions" message is your evidence of parallel planes or a line of intersection: say so in words.
Linking questionVectors turn the geometry of Trigonometry: triangles into algebra (the angle formula is the cosine rule in disguise). Velocity vectors link to Kinematics (Integration and kinematics), and complex numbers behave like 2D vectors under addition (Complex numbers).
Idea for your IA / explorationUse vectors to model flight paths or drone routes in 3D and find the closest approach between two moving objects, or explore computer graphics: how 3D scenes are projected onto a 2D screen.

Test yourself — cover the answers first

Quick check

  1. HL Find |(2, −3, 6)|.
  2. HL Find (1, 2, 3) · (4, −1, 0).
  3. HL What does a zero dot product tell you?
  4. HL Give a normal vector to the plane 2x − y + 5z = 7.
  5. HL What are skew lines?
  6. HL Is (1, 0, 0) × (0, 1, 0) equal to k or −k?

Exam-style

  1. HL Show that L1: r = (1, 0, 0) + λ(1, 1, 0) and L2: r = (0, 1, 3) + μ(0, 1, 1) are skew.
  2. HL Find the acute angle between the planes 5x + 3y + 4z = 13 and x + y + z = 6.
Show answers
  1. 7
  2. 2
  3. the vectors are perpendicular
  4. (2, −1, 5)
  5. lines that are not parallel and do not intersect
  6. k
  7. not parallel: (1, 1, 0) is not a multiple of (0, 1, 1)
    x: 1 + λ = 0 ⇒ λ = −1; y: −1 = 1 + μ ⇒ μ = −2
    z: 0 = 3 + (−2) = 1 ✗, no common point, so skew AG
  8. normals (5, 3, 4) and (1, 1, 1): dot product 12
    cos θ = 12√50 √3
    θ = 11.5°
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.