Some quantities in physics need only a size to be fully described — a temperature of 20 °C, a mass of 3 kg. Others make no sense until you also say which way — a push of 10 N means nothing until you know the direction it points. That single distinction, size-only versus size-plus-direction, splits every physical quantity into two camps: scalars and vectors. Getting it right is the foundation for the whole of mechanics.
📘 What you need to know
A scalar has magnitude only (e.g. mass, time, energy)
A vector has magnitude and direction (e.g. weight, velocity, force)
Distance is a scalar (total path length); displacement is a vector (straight line from start to finish, with direction)
Speed is a scalar; velocity is a vector — you can travel at constant speed yet have changing velocity if you turn
Vectors are drawn as arrows: length shows magnitude, the arrowhead shows direction
A vector can be split into components — usually a horizontal Fx and vertical Fy
A handy test: if a quantity can sensibly be negative, it’s probably a vector
Scalars and vectors
A scalar is a quantity that is fully described by its magnitude alone. Mass is a scalar: 3 kg is 3 kg, there’s no “3 kg to the left”. A vector carries both a magnitude and a direction. Weight is a vector because it’s a force — it has a size (in newtons) and always points downwards, towards the centre of the Earth.
Here’s a quick gut-check I give students: ask whether the quantity can be negative. Can you have −5 J of energy? No — energy is a scalar. Can you have −5 m of displacement? Absolutely, it just means 5 m in the opposite direction. If a minus sign makes physical sense, you’re almost certainly dealing with a vector.
Distance versus displacement
Distance is how far an object travels along its actual route — the total length of the path, twists and turns included. It has size but no direction, so it’s a scalar. Displacement is different: it’s the straight-line gap from the starting point to the finishing point, together with the direction of that line. Because it carries a direction, displacement is a vector.
Distance is the length of the winding route; displacement is the straight arrow from start to finish, and it points.
Speed versus velocity
The same split runs through motion. Speed is the distance travelled per unit time — it tells you how fast, but not which way, so it’s a scalar. Velocity is the displacement per unit time; it tells you how fast and in what direction, making it a vector.
A car going round a roundabout at a steady 30 km h⁻¹ has constant speed but changing velocity — because the direction of travel keeps changing, even though the speedometer never moves. That’s the whole idea behind circular motion: constant speed, forever-changing velocity.
Examples of scalars and vectors
It’s worth committing the common ones to memory — they come up again and again, and a single mislabel can wreck a whole answer.
Scalars (magnitude only)
Vectors (magnitude & direction)
distance
displacement
speed
velocity
mass
acceleration
time
force
energy
momentum
volume, density, pressure
weight
electric charge, temperature
—
Representing vectors as arrows
Because a vector carries two pieces of information, we draw it as an arrow. The length of the arrow (to a chosen scale) shows the magnitude, and the arrowhead shows the direction. A longer arrow means a bigger quantity; rotate the arrow and you’ve changed the direction.
An arrow can also be broken into components — two perpendicular arrows that, added together, have exactly the same effect as the original. We usually split into a horizontal part Fx and a vertical part Fy, drawn as dotted lines.
A force F resolved into a horizontal component Fx and a vertical component Fy. Together they replace F.
The trig ratios do the splitting: the horizontal (adjacent) side uses cosine, the vertical (opposite) side uses sine.
Resolving a vectorFx = F cos θ · Fy = F sin θ
Quick recap: scalars have size only, vectors have size and direction; distance/speed/mass/energy are scalars, displacement/velocity/force/momentum are vectors; draw vectors as arrows and split them with cos (horizontal) and sin (vertical).
WE 1
A runner jogs 400 m around a circular track and stops exactly where they started. State the distance travelled and the displacement, and explain the difference.
Distance
= total path length = 400 m
it’s a scalar, so only the size matters
Displacement
start point and finish point are the same
straight-line gap between them = 0
distance = 400 m, displacement = 0 mDisplacement is zero because the runner ends where they began — direction and start-to-finish position are what count for a vector.
WE 2
A child pulls a sledge with a rope. The tension in the rope is 60 N, directed at 25° above the horizontal. Calculate the horizontal and vertical components of this force.
Resolve the 60 N force using cosine for the horizontal part and sine for the vertical part.
Horizontal component
Fx = F cos θ
Fx = 60 × cos 25°Fx = 54 NVertical component
Fy = F sin θ
Fy = 60 × sin 25°Fy = 25 NThe horizontal part drags the sledge forward; the vertical part lifts it slightly, easing the load on the ground.
💡 Top tips
The minus-sign test settles most cases fast: if negative is meaningful, it’s a vector
“cos SANDWICH”: cosine always goes with the adjacent side, so the horizontal component (along the angle’s base) uses cos θ
Distance ≥ magnitude of displacement, always — and they’re only equal for motion in a straight line
Constant speed doesn’t mean constant velocity. Turning changes velocity even at fixed speed
⚠ Common mistakes
Treating distance and displacement (or speed and velocity) as the same thing — the vector always carries direction
Calling acceleration or momentum scalars — both are vectors
Swapping sin and cos when resolving — check which side of the triangle you’re finding first
Forgetting the calculator must be in degrees for angles given in degrees
Up next: Combining & Resolving Vectors — now that you can draw an arrow and split it into components, we’ll add and subtract vectors with the triangle and parallelogram methods, and find the resultant.
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