IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~5 min read HL only

Introduction to Vectors

A scalar has size only (mass, time, speed). A vector has size and direction (velocity, force, displacement). Vectors can be written three ways — as bold/underlined letters (a), as columns, or in i, j, k base form. Same object, three notations.

📘 What you need to know

Scalar: size only (e.g., 5 kg, 3 seconds). Vector: size + direction (e.g., 5 m/s east).
Three notations for the same vector: bold/underlined a; column (x, y, z); base form xi + yj + zk.
Base vectors: i = (1, 0, 0); j = (0, 1, 0); k = (0, 0, 1) — unit vectors along the x, y, z axes.
Arrow notation: vector from A to B is written AB.
Two vectors are equal ⟺ all corresponding components match.
Components can be positive or negative — sign tells you which direction along the axis.
Vectors can be 2D or 3D — IB AA HL uses both.

Scalar or vector?

Scalar

size only

mass, time, distance, temperature, speed

Vector

size + direction

velocity, force, displacement, acceleration, momentum

“Speed 60 km/h” is a scalar. “Velocity 60 km/h heading east” is a vector — same number, but the direction makes it a vector.

Three ways to write the same vector

All three describe the same vector a = 2−35 = 2i − 3j + 5k

The base vectors i, j, k are unit vectors along the positive x, y, z axes. Any 3D vector splits into a sum of multiples of them.

Missing components: when going from base form to column, fill in zeros for any missing axis. k − 2j = 0i − 2j + k = column (0, −2, 1).

When are two vectors equal?

Two vectors are equal if and only if every corresponding component is equal. So if (a, b, c) = (3, −2, 7), then a = 3, b = −2, c = 7 — and that’s the only way.

🧭 Recipe — switch between column and base vector form

Identify the components: x (the i coefficient), y (the j coefficient), z (the k coefficient).
Insert zeros for missing terms. i − 4k means y = 0 → column (1, 0, −4).
Watch the signs. −3j means y-component is −3, not 3.
Drop zero terms when going to base form. (5, 0, −2) = 5i − 2k.
Check by counting components: a 3D vector has exactly three numbers in its column.

Worked examples

WE 1

Classify each as scalar or vector

State whether each of the following is a scalar or a vector quantity:
(a) A train moves at 90 km/h heading southwest.
(b) A book has a mass of 1.4 kg.
(c) A swimmer is displaced 25 m due north from her starting position.
(d) A kettle takes 4 minutes to boil.

(a) Speed + direction → vector (velocity) (a) Vector (b) Mass has no direction (b) Scalar (c) Displacement always has direction (c) Vector (d) Time has no direction (d) Scalar

WE 2

Column → base vector form

Write the vector with column form (5, −2, 4) using base vector notation.

Step 1: Identify components x = 5, y = −2, z = 4 Step 2: Plug into xi + yj + zk 5i + (−2)j + 4k 5i − 2j + 4k

WE 3

Base vector → column form (with missing term)

Write the vector 4i − k using column vector notation.

Step 1: Identify each component (j is missing → coefficient 0) 4i − k = 4i + 0j − 1k Step 2: Read off (x, y, z) = (4, 0, −1) 40−1 missing term in base form means a 0 in the column — never leave a slot blank

WE 4

Equal vectors → find unknowns

Given that (a − 1, 2b, 5) and (3, −8, c + 1) represent the same vector, find a, b, and c.

Equal vectors → corresponding components equal Step 1: x-component a − 1 = 3 → a = 4 Step 2: y-component 2b = −8 → b = −4 Step 3: z-component c + 1 = 5 → c = 4 a = 4, b = −4, c = 4

WE 5

Describe what the vector represents

Describe the movement represented by the vector −3i + 7j − 2k in 3D space.

Read each component as a signed displacement along its axis −3i → 3 units in the negative x direction +7j → 7 units in the positive y direction −2k → 2 units in the negative z direction 3 left, 7 up (y), 2 in (negative z) the sign attached to i, j, k is which side of the origin you go on each axis

WE 6

Build a vector from a description

A point moves 4 units in the positive x direction, 0 units in the y direction, and 6 units in the negative z direction. Write the displacement as (a) a column vector and (b) in base vector form.

Step 1: Read the components with correct signs x = 4, y = 0, z = −6 (a) Column form (a) 40−6 (b) Base form (drop the 0j term) (b) 4i − 6k

💡 Top tips

Pick one notation and stick with it for a question — column or base form, whichever feels cleaner.
Always include zeros in column form. (1, 0, −4), not (1, −4) — that would be a 2D vector.
Direction is a key word for spotting vectors. “Moves at 5 m/s north” = vector; “5 m/s” alone = scalar.
Equal vectors means every component matches — set up one equation per component.
Watch signs carefully when converting between notations — “−4k” is a z-component of −4, not 4.

⚠ Common mistakes

Treating speed as a vector. Speed is scalar (just a number); velocity adds direction.
Skipping the zero in column form when a base term is missing. i − 2k = (1, 0, −2), not (1, −2).
Confusing inverse and reciprocal trig with vector notation: i, j, k are vectors, not real numbers.
Wrong axis for k. i goes along x, j along y, k along z — alphabetical order matches axis order.
Mixing 2D and 3D. (3, −2) is a 2D vector; (3, −2, 0) is its 3D version with z = 0.

Next note: Parallel Vectors. Two vectors point the same way (or directly opposite) if and only if one is a scalar multiple of the other — a condition that turns up everywhere in vector geometry.

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