IB Physics HLTool 3 — MathematicsPractical Skillsmagnitude vs direction~10 min read
Scalars & Vectors
Every quantity in physics is one of two types: a scalar or a vector. It sounds like a small distinction, but it quietly decides how you add quantities, whether a minus sign matters, and half the mistakes students make. Get this sorted early and everything downstream — forces, motion, momentum, fields — becomes much easier to reason about.
📚 What you need to know
A scalar has magnitude only (a size, no direction)
A vector has magnitude and direction
Distance and speed are scalars; displacement and velocity are vectors
Vectors are drawn as arrows: length = magnitude, arrowhead = direction
A quantity that can be negative (point the other way) is usually a vector
Vectors can be split into perpendicular components
Scalars
A scalar is a quantity that has magnitude only — a number and a unit, but no direction attached. If someone tells you the value, you know everything there is to know about it.
For example, mass is a scalar. A bag has a mass of 5 kg — there’s no such thing as “5 kg to the left”. Other scalars include distance, speed, time, energy, volume, density, and temperature.
Vectors
A vector is a quantity that has both magnitude and direction. Knowing the size alone isn’t enough — you also need to know which way it points.
For example, weight is a vector: it’s a force with a size (in newtons) and a direction (downwards, towards the Earth). Other vectors include displacement, velocity, acceleration, force, and momentum.
Here’s the quick gut-check I give every student: ask “can this quantity be negative, meaning it points the opposite way?” You can have negative displacement (backwards) or negative velocity (reversing) — those are vectors. You can’t have negative mass or negative energy — those are scalars. It’s not a formal rule, but it catches the answer nearly every time.
Distance vs displacement, speed vs velocity
The cleanest way to see the scalar–vector split is to compare the pairs that get confused most.
Distance is the total length of the path travelled, regardless of direction — a scalar.
Displacement is the straight-line distance and direction from start to finish — a vector.
Speed is distance per unit time, no direction — a scalar.
Velocity is displacement per unit time, including direction — a vector.
This is why an object can travel at constant speed but changing velocity: think of a car going round a roundabout at a steady 30 km h−1. The speed never changes, but the direction constantly does — so the velocity is always changing.
A handy table
Here are the common quantities sorted into their two camps — worth memorising, because exam questions love to test the boundary cases.
Scalars (magnitude only)
Vectors (magnitude + direction)
distance
displacement
speed
velocity
mass
acceleration
time
force
energy
momentum
volume, density
weight
temperature, charge
—
Representing vectors as arrows
Because a vector carries direction, we draw it as an arrow. Two things about that arrow matter:
The length of the arrow is proportional to the magnitude — a longer arrow means a bigger quantity.
The arrowhead shows the direction the vector points.
Both arrows point the same way, but the 6 N arrow is drawn twice as long as the 3 N arrow to show its larger magnitude.
Sometimes a vector is drawn with a small arrow above its symbol, or split into a horizontal component (Fx) and a vertical component (Fy) using dotted lines. Splitting a vector into perpendicular parts like this is called resolving — it gets its own page.
WE 1
A runner jogs 400 m around a circular track and finishes exactly where they started. State their distance and their displacement, and say which is a scalar and which is a vector.
Distance
The total path length = 400 m — a scalar.
Displacement
Start and finish are the same point, so straight-line distance = 0 m — a vector.
Distance 400 m (scalar); displacement 0 m (vector)This is the classic case that shows why the two aren’t the same — you can travel a long distance yet end with zero displacement.
WE 2
Classify each quantity as a scalar or a vector: (a) temperature, (b) acceleration, (c) energy, (d) momentum.
Apply the “can it be negative-as-direction?” test
(a) temperature → scalar
(b) acceleration → vector
(c) energy → scalar
(d) momentum → vectorscalar, vector, scalar, vectorAcceleration and momentum both need a direction to be fully described, so they’re vectors. Temperature and energy are pure sizes.
💡 Top tips
Scalar = size only; vector = size + direction.
Distance/speed are scalars; displacement/velocity are vectors.
Constant speed can still mean changing velocity (if direction changes).
Draw vectors to scale: length = magnitude, arrowhead = direction.
Use the “can it be negative?” gut-check when unsure.
⚠ Common mistakes
Calling speed a vector — it has no direction, so it’s a scalar
Thinking distance and displacement are always equal (only true in a straight line)
Forgetting that turning at constant speed changes the velocity
Giving a vector answer with no direction
Assuming energy or mass can be negative
Quick recap: A scalar has magnitude only (mass, speed, energy, temperature); a vector has magnitude and direction (displacement, velocity, force, momentum). Vectors are drawn as arrows — length for size, arrowhead for direction — and can be split into perpendicular components.
Now that you can tell scalars and vectors apart, the natural next step is learning what to do with vectors when several act at once. Adding them, and splitting them into components, is exactly what comes next in Adding & Resolving Vectors.
Scalars and vectors still blurring together?
Book a free meeting and we’ll nail the difference with quick examples, so you never lose an easy mark mixing up speed and velocity or forgetting a direction again.
