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IB DP Maths AI · Topic 3 Geometry and trigonometry · AHL 3.7–3.13 · ~8 min read

Radians, vectors and matrix transformations

HL geometry for Maths AI: radians, arcs and sectors, the unit circle and Pythagorean identity, solving trig equations, geometric transformations with 2 × 2 matrices, vectors and the scalar product, vector equations of lines, vector kinematics with closest approach, and the vector product.

RadiansArc length & sector areaUnit circlesin²θ + cos²θ = 1Trig equationsMatrix transformationsComposite transformationsDeterminant as area factor
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

  • Radians: π rad = 180°. With θ in radians, arc length l = rθ and sector area A = ½r2θ.
  • Unit circle: the point at angle θ is (cos θ, sin θ), so sin2θ + cos2θ = 1 and tan θ = sin θcos θ.
  • Matrix transformations: a 2 × 2 matrix maps (x; y) to a new point. ǀdetǀ is the area scale factor. For "A then B" the matrix is BA.
  • Vectors: v · w = ǀvǀǀwǀ cos θ gives angles; r = a + λb is a line; r = r0 + tv describes constant-velocity motion, with speed ǀvǀ.
  • This note is HL only. The vector product section leaves the course in 2029 (see the Syllabus note).

By the end you can

  • Use radians, including arc length and sector area. HL
  • Use the unit circle, sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ, and solve trig equations in a given interval. HL
  • Describe and apply transformations with matrices, including composites, and use the determinant as an area scale factor. HL
  • Use vectors, the scalar product and the angle between vectors. HL
  • Write vector equations of lines and use them in kinematics, including closest approach. HL
  • Find and use the vector product. HL
Syllabus noteFrom first exams in 2029, the vector (cross) product leaves the AI HL syllabus. Everything else here stays. If you sit exams up to November 2028, you still need the vector product too.

1Radians and the unit circle HL

π/6π/4π/3π/25π/6π3π/2(√3/2, ½)π rad = 180°x = cos θy = sin θsin²θ + cos²θ = 1Radians on the unit circle: one full turn is 2π.
Learn the key angles in radians and their coordinates on the unit circle.
l = rθ, A = ½r2θ FORMULA BOOKLET
θ in radians
Worked example

Convert 150° to radians, and find the arc length of a sector of radius 8 cm and angle 1.2 rad.

  1. 150 × π180 = 5π6 A1
  2. l = 8 × 1.2 = 9.6 cm A1
cos2θ + sin2θ = 1, tan θ = sin θcos θ FORMULA BOOKLET
identities
Worked example

θ is obtuse and sin θ = 0.6. Find cos θ and tan θ.

  1. cos2θ = 1 − 0.36 = 0.64, and cos θ < 0 for obtuse θ M1R1
  2. cos θ = −0.8, tan θ = −0.75 A1
Worked example

Solve 2 sin x = 1 for 0 ≤ x ≤ 2π.

  1. sin x = 0.5 M1
  2. x = π6 and π − π6 M1
  3. x = π6, 5π6 (or 0.524, 2.62) A1
GDC (calculator) methodSet the GDC to radians, graph y = 2 sin x and y = 1 on 0 ≤ x ≤ 2π, and use intersect. Check the window matches the interval exactly.

2Matrix transformations HL

TransformationMatrix
rotation θ anticlockwise about O(cos θ −sin θ; sin θ cos θ)
reflection in y = (tan θ)x(cos 2θ sin 2θ; sin 2θ −cos 2θ)
enlargement, scale factor k, centre O(k 0; 0 k)
horizontal stretch, factor k(k 0; 0 1)
vertical stretch, factor k(1 0; 0 k)
  • The rotation and reflection matrices are in the formula booklet. For any other transformation, see where (1; 0) and (0; 1) go: those images are the two columns of the matrix. So the rotation by 90° is (0 −1; 1 0), sending (1; 0) to (0; 1).
xyobjectimage(3, 1)(−1, 3)rotation 90°anticlockwise:(0 −1; 1 0)det = 1: same area
The rotation matrix moves each vertex; the shape keeps its area because det = 1.
Worked example

The triangle with vertices (1, 0), (3, 0), (3, 1) is rotated 90° anticlockwise about the origin. Find the image of (3, 1).

  1. (0 −1; 1 0)(3; 1) M1
  2. (−1; 3) A1
  • Composite: "transformation A followed by B" has matrix BA (the first one goes next to the vector).
  • Area: area of image = ǀdet Mǀ × area of object. A negative determinant means the shape has been reflected.
Worked example

A shape of area 4 cm2 is transformed by M = (3 1; 1 2). Find the area of the image.

  1. det M = 6 − 1 = 5 A1
  2. area = 5 × 4 = 20 cm2 A1
Common mistakeMultiplying in the wrong order for composites. "Reflect, then rotate" is (rotation)(reflection), not the other way round.

3Vectors and the scalar product HL

ǀvǀ = √(v12 + v22 + v32) FORMULA BOOKLET
magnitude
v · w = v1w1 + v2w2 + v3w3 = ǀvǀǀwǀ cos θ FORMULA BOOKLET
scalar product; 0 means perpendicular
Worked example

Find the angle between a = (3, 0, 4) and b = (2, 2, 1).

  1. a · b = 6 + 0 + 4 = 10, ǀaǀ = 5, ǀbǀ = 3 A1
  2. cos θ = 1015 M1
  3. θ = 48.2° A1
  • Position vectors: OA is the vector from the origin to A, and AB = b − a.

4Vector equations and kinematics HL

r = a + λb FORMULA BOOKLET
a line through a with direction b
r = r0 + tv
position at time t; v is the velocity, ǀvǀ the speed
x (km)y (km)024681012246A at t = 0: (2, 3)B at t = 0: (10, 0)closest at t = 1.65 hr_A = (2, 3) + t(4, 1)r_B = (10, 0) + t(−1, 2)min distance 1.37 km
Two ships on straight courses. Their distance apart is smallest when the line joining them is perpendicular to their relative velocity.
Worked example

Ship A is at (2, 3) + t(4, 1) km and ship B at (10, 0) + t(−1, 2) km, t in hours. Find the speed of A and the closest distance between the ships.

  1. speed of A = ǀ(4, 1)ǀ = √17 = 4.12 km h−1 A1
  2. B − A = (8 − 5t, −3 + t) M1
  3. d2 = (8 − 5t)2 + (t − 3)2; minimum (GDC or calculus) at t = 4326 = 1.65 h M1A1
  4. closest distance 1.37 km A1
  • Do they collide? Only if they are at the same point at the same time t. Paths that cross at different times are not a collision.

5The vector product HL

v × w = (v2w3 − v3w2, v3w1 − v1w3, v1w2 − v2w1), ǀv × wǀ = ǀvǀǀwǀ sin θ FORMULA BOOKLET
perpendicular to both v and w
Worked example

Find (1, 2, 0) × (0, 1, 3), and the area of the parallelogram these vectors make.

  1. (2 × 3 − 0 × 1, 0 × 0 − 1 × 3, 1 × 1 − 2 × 0) M1
  2. = (6, −3, 1) A1
  3. area = √(36 + 9 + 1) = √46 ≈ 6.78 A1
Linking questionRadians and sinusoidal models connect to Modelling with functions; rotation matrices and complex numbers both rotate the plane (Complex numbers); velocity vectors return in Differential equations and phase portraits. For lines and planes in 3D with more algebra, see Maths AA Vectors, lines and planes.
Idea for your IA / explorationModel the flight paths of two aircraft or drones with vector equations and find their closest approach, or use matrix transformations to explain how a phone app rotates and resizes photos.

Test yourself — cover the answers first

Quick check

  1. HL Convert 2 rad to degrees.
  2. HL Find the sector area for r = 4 and θ = 0.5 rad.
  3. HL Write the matrix for an enlargement with scale factor 3 about O.
  4. HL det M = −2. What happens to the area and orientation?
  5. HL Find ǀ(2, −3, 6)ǀ.
  6. HL What does v · w = 0 mean?

Exam-style

  1. HL Find the matrix of a reflection in y = x followed by a rotation of 90° anticlockwise about O, and describe the single transformation.
  2. HL A particle moves with r = (1, 4) + t(3, −2). Find when it crosses the x-axis and its speed.
Show answers
  1. 115°
  2. 4
  3. (3 0; 0 3)
  4. area doubles; the shape is reflected
  5. 7
  6. v and w are perpendicular
  7. reflection (0 1; 1 0), rotation (0 −1; 1 0)
    combined = (0 −1; 1 0)(0 1; 1 0) = (−1 0; 0 1)
    a reflection in the y-axis
  8. y = 4 − 2t = 0 ⇒ t = 2 (at (7, 0))
    speed = √(9 + 4) = √13 ≈ 3.61
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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.