IB Maths AI HLVector PropertiesPaper 1 & 2Scalars & vectors~7 min read
Introduction to Vectors
A scalar is a quantity with magnitude only — mass, time, distance, speed. A vector has magnitude and direction — velocity, force, displacement, acceleration. Vectors are written in two equivalent forms: base-vector notation using i, j, k (the unit vectors along the three axes), or column-vector notation stacking the components vertically. Knowing how to read, convert, and combine these forms is the foundation of every vector question that follows.
📘 What you need to know
Scalar vs vector: scalars have magnitude only (mass, time, distance, speed, temperature); vectors carry direction too (displacement, velocity, acceleration, force, momentum).
Bold or arrow notation: write a vector as a, a, or with an arrow above two endpoints, e.g. AB with an arrow over it = vector from A to B.
Base vectors: i = (1, 0, 0)T along +x, j = (0, 1, 0)T along +y, k = (0, 0, 1)T along +z. Each has magnitude 1.
Any 3D vector can be written as xi + yj + zk, which is identical to the column (x, y, z)T.
Equal vectors: two vectors are equal if and only if every corresponding component matches.
Both notations are acceptable in your exam — use whichever is faster for the calculation in front of you.
Two notations — one vector
Every vector can be written two equivalent ways: stacked vertically as a column, or as a sum of base vectors with each component multiplying its corresponding axis unit. The information is identical; the form is a matter of convenience. Column form is cleaner for adding and subtracting; base-vector form reads more naturally in prose. Switching freely between the two is a basic skill the IB expects on every vector question.
The vector v = 4i + 3j from O to (4, 3) breaks into a horizontal step of 4 units along i and a vertical step of 3 units along j. The same idea extends to 3D with a third component along k.
Column form ⇔ Base vector form
(x, y, z)T = xi + yj + zktwo equal vectors share every component; missing terms (e.g. j not written) correspond to a 0 in that row of the column form
Identifying scalars vs vectors
A quantity is a vector if its description includes a direction (a bearing, “forward”, “downward”, “north”); it’s a scalar if only the magnitude is given. Mass, time, speed, distance, and temperature are scalars. Displacement, velocity, acceleration, force, and momentum are vectors. The distinction matters because vectors can have negative components, while scalars like mass or time cannot. Watch the language carefully — “speed” is a scalar, “velocity” is a vector, even though both have the same units.
Quick test: ask “could this quantity be reversed?” If reversing makes physical sense (north vs south, up vs down, forward vs backward), it’s a vector. If reversing is meaningless (a duration of −5 minutes? a mass of −2 kg?), it’s a scalar.
🧠Recipe — convert between forms
Column → base vector: read each row (top, middle, bottom) as the coefficient of i, j, k. Omit zero terms.
Base vector → column: place each coefficient in the matching row — i on top, j in the middle, k at the bottom. Insert 0 for missing components.
Mind the order: re-arrange “3i + k − 7j” mentally into i, j, k order before converting.
For scalar-vector decisions: ask whether a direction (bearing, “up”, “right”) is part of the quantity.
For equal vectors: set corresponding components equal, then solve component-by-component.
Worked examples
WE 1
Identify scalar or vector
State whether each of the following is a scalar or a vector quantity:
(a) A drone flies at 12 m/s on a bearing of 215°.
(b) A bag of rice has mass 4.5 kg.
(c) An aircraft accelerates forward at 3.2 m/s2.
(d) A bus journey takes 47 minutes.
(e) A box experiences a 25 N force vertically downward.
(f) A swimmer covers a distance of 1500 m.
test: “does it carry a direction?”(a) velocity — speed + bearing → vector(b) mass — no direction → scalar(c) acceleration + “forward” → vector(d) time — no direction → scalar(e) force + “downward” → vector(f) distance (magnitude only) → scalarvectors: (a), (c), (e); scalars: (b), (d), (f)“speed” alone is scalar; “velocity” with direction is vector. Same units, different category.
WE 2
Convert column form to base vector form
Write the vector (5, −2, 0)T in base vector (i, j, k) notation.
read each row as a coefficient5 → coefficient of i−2 → coefficient of j0 → coefficient of k (omit this term)5i − 2ja zero in any row means that base vector drops out of the sum.
WE 3
Convert base vector form to column form (out-of-order)
Write the vector 3i + k − 7j as a column vector.
re-order into i, j, k3i − 7j + kplace each coefficient in its rowi-coefficient: 3 (top)j-coefficient: −7 (middle)k-coefficient: 1 (bottom — k means 1k)(3, −7, 1)Talways sort into i, j, k order before reading off the column — order of writing doesn’t change the vector.
WE 4
Simplify a vector expression
Simplify the expression 2(i − j + 3k) − (i + 2k), giving your answer in (a) base vector form, (b) column vector form.
distribute the scalar 22(i − j + 3k) = 2i − 2j + 6ksubtract the second vector(2i − 2j + 6k) − (i + 2k)= (2−1)i + (−2−0)j + (6−2)k= i − 2j + 4k(a) i − 2j + 4k; (b) (1, −2, 4)Tmissing j term in (i + 2k) acts as 0j when subtracting.
WE 5
Equal vectors — find the unknowns
The two vectors u = (2a − 1, b + 3, 5)T and v = (7, 1, c − 2)T are equal. Find the values of a, b, and c.
vectors equal ⇒ each component equal2a − 1 = 7 → 2a = 8 → a = 4b + 3 = 1 → b = −25 = c − 2 → c = 7a = 4, b = −2, c = 7three independent equations from one vector equality — standard exam pattern.
WE 6
Write a vector from a description
A particle moves 3 units in the positive x direction, 1 unit in the negative y direction, and 4 units in the positive z direction. Write the displacement vector in (a) base vector form, (b) column vector form.
identify each axis component (with sign)x: +3 → 3iy: −1 → −j (negative direction)z: +4 → 4kcombine into one vector(a) 3i − j + 4k; (b) (3, −1, 4)T“negative y direction” means coefficient is negative; “1 unit” of j is written as just −j, not −1j.
💡 Top tips
Look for direction words: bearing, north, upward, forward, downward — any of these promote a quantity from scalar to vector.
Use whichever notation is faster: column for adding/subtracting many components, base-vector for writing answers in words.
“1i” is written as just i; “−1j” as just “−j“. Same for k.
Missing terms are zeros: if the j coefficient isn’t written, it’s 0 — the column form must still show that 0.
Re-order before converting: a vector written as 5k + 2i − 3j goes into column form as (2, −3, 5)T, not (5, 2, −3)T.
âš Common mistakes
Confusing speed with velocity — speed is scalar, velocity is the same magnitude with an added direction.
Missing-component traps: writing 2i + 3k as (2, 3, 0)T instead of (2, 0, 3)T.
Order errors: components placed in the order the question wrote them, not in i–j–k order.
Sign errors: missing the minus sign when a vector points in the negative direction of an axis.
Treating equal vectors loosely — all three components must match, not just one or two.
Next up — Parallel Vectors. Two vectors point in the same (or exactly opposite) direction if one is a scalar multiple of the other. This single test — finding the scalar k such that a = kb — underlies dozens of exam questions on parallelism, ratios, and proportional sides of a shape.
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