Mini-Lesson

Geometry & Trigonometry (HL)

This HL mini-lesson adds the HL extensions to the SL geometry toolkit: vectors in 2D and 3D (magnitude, the scalar (dot) product and the angle between vectors), and matrix transformations of the plane (rotations, reflections, enlargements), alongside the sine/cosine rules and 3D work.

AI HL flavour: vectors describe forces and directions; matrices encode geometric transformations used in graphics and robotics.

Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Every number on the calculation screens has been re-derived and checked. Press Start when you are ready.

Vectors

Vectors & magnitude

A vector has magnitude and direction. In components a = (a₁, a₂) or (a₁, a₂, a₃):

|a| = √(a₁² + a₂²)3D: |a| = √(a₁² + a₂² + a₃²)

Worked example

a = (3, 4): |a| = √(9 + 16) = 5.

3D vector (2, 3, 6): |v| = √(4 + 9 + 36) = √49 = 7

Calculate

Your turn — magnitude

1Find the magnitude of the vector a = (3, 4).

Hint: √(3² + 4²).

Calculate

Your turn — 3D magnitude

2Find the magnitude of the 3D vector v = (2, 3, 6).

Hint: √(2² + 3² + 6²) = √49.

Scalar product

The dot product & angle between vectors

The scalar (dot) product measures alignment:

a·b = a₁b₁ + a₂b₂a·b = |a||b| cos θ

Worked example

a = (3, 4), b = (2, −1).

a·b = 3×2 + 4×(−1) = 6 − 4 = 2

|a| = 5, |b| = √5 = 2.236

cos θ = 2 ⁄ (5 × 2.236) ⇒ θ = 79.70°

If a·b = 0 the vectors are perpendicular.

Calculate

Your turn — dot product

3Find the scalar (dot) product a·b for a = (3, 4) and b = (2, −1).

Hint: 3×2 + 4×(−1).

Calculate

Your turn — angle between vectors

4Find the angle θ (in degrees, 2 d.p.) between a = (3, 4) and b = (2, −1). Use cos θ = (a·b) ⁄ (|a||b|).

°

Hint: a·b = 2, |a| = 5, |b| = √5; cos θ = 2 ⁄ (5√5).

Quick check

Perpendicular vectors

?Two non-zero vectors are perpendicular exactly when their scalar product is:

Matrix transformations

Transforming the plane with matrices

A 2×2 matrix maps each point (x, y) to a new point. Common transformations:

Rotation by θ about O: [[cos θ, −sin θ], [sin θ, cos θ]] (det = 1).
Reflection in the x-axis: [[1, 0], [0, −1]] (det = −1).
Enlargement scale k about O: [[k, 0], [0, k]] (det = k²).

The determinant gives the area scale factor; a negative determinant means orientation is reversed (a reflection).

Non-right triangles

Sine & cosine rules (still assessed)

HL still uses the sine and cosine rules and the area formula.

c² = a² + b² − 2ab cos CArea = ½ ab sin C

Worked example

a = 7, b = 10, C = 45°.

c = √(149 − 98.99) = 7.07; Area = ½×7×10×sin45° = 24.75

Calculate

Your turn — cosine rule

5A triangle has a = 7, b = 10 and included angle C = 45°. Find side c to 2 decimal places.

Hint: c² = 7² + 10² − 2×7×10×cos45°.

Sort it

Geometry HL sort

Tap an object, then the tool it belongs to.

➡️ Vector operation

🔄 Matrix transformation

△ Triangle rule

Quick check

Determinant of a transformation

?A 2×2 transformation matrix has determinant −1. Geometrically this is a:

Quick check

Dot product sign

?If a·b is negative, the angle between the vectors is:

Match it

Match the geometry tool to its result

Tap an item on the left, then its match on the right.

Item

Match

Vector equation of a line

Lines in vector form

A line can be written r = a + td, where a is a point on it and d its direction. Varying the parameter t traces the whole line.

Two lines are parallel when their direction vectors are scalar multiples of each other.

Unit vectors

Unit vectors & direction

A unit vector has magnitude 1. To get the unit vector in the direction of a, divide by its magnitude: â = a ⁄ |a|.

Worked example

a = (3, 4), |a| = 5, so â = (0.6, 0.8), which has magnitude 1.

Quick check

Direction vectors

?Two lines have direction vectors (2, 4) and (1, 2). The lines are:

Recap

The big ideas to take away

Vector magnitude: |a| = √(a₁²+a₂²(+a₃²))

Dot product: a·b = a₁b₁+a₂b₂ = |a||b|cos θ; 0 ⇒ perpendicular

Angle between: cos θ = (a·b) ⁄ (|a||b|)

Matrix transforms: rotation, reflection, enlargement; det = area factor

Sine/cosine rules: still assessed for any triangle

Area: ½ ab sin C

You have worked through the whole topic. Press Finish to see your score.

🏆

Mini-lesson complete!

⭐⭐⭐

You have covered HL Geometry & Trigonometry (vectors and transformations) for AI. 🎉

Your stars: 0 / 0

Next: test yourself in the Evaluate stage, then lock it in with Verify.

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