Imagine you are guiding a rescue helicopter by radio, and all you can tell the pilot is: "They're 5 km from home." The pilot still has no idea where to fly. 5 km north? 5 km out to sea? The distance alone is useless — a circle of possible spots 10 km wide. What the pilot actually needs is 5 km, on a bearing of north-east: a size and a direction. Only then does one number point at one place.

That is the whole idea of this page. Physics splits every measurable quantity into two families. Some quantities are complete with just a size — "2 kg", "40 °C", "3 hours". Others are only half-told until you add a direction — "5 km north", "300 m/s upwards", "20 N to the left". The first family are scalars; the second are vectors. Getting them straight is the key that unlocks velocity and acceleration, forces and momentum — because those are all vectors, and vectors don't add up the way plain numbers do.

Two families: scalars and vectors

Here are the two definitions, as plainly as they can be put:

• A scalar has only a magnitude (a size, a number with a unit). There is no arrow to draw — the quantity is finished as soon as you say how much.
• A vector has a magnitude and a direction. Miss out the direction and you have only told half the story.

The GCSE quantities you need to sort into these two boxes are:

• Scalars (size only): distance, speed, mass, time, energy, temperature.
• Vectors (size + direction): displacement, velocity, acceleration, force (including weight), momentum.

Notice the neat pairing that runs down the two lists: distance pairs with displacement, and speed pairs with velocity. In each pair the scalar is "how much", and the vector is "how much, and which way". That pairing is worth a card of its own — but first, how do we draw a vector?

Drawing a vector as an arrow

A vector is perfectly captured by a single arrow, and the arrow does two jobs at once:

• its length stands for the magnitude — a longer arrow means a bigger quantity, drawn to a chosen scale (say 1 cm on paper = 1 N of force);
• the way it points — marked by the arrowhead — stands for the direction.

So a force of 10\ \text{N} to the right is drawn twice as long as a force of 5\ \text{N} to the right, and a force of the same 10\ \text{N} pointing up is the same length but turned a quarter-circle. Two arrows are the same vector only if they have both the same length and the same direction — change either one and you have a different vector. This is exactly the free-body arrows you drew for resultant force: every force is a vector, so every force is an arrow.

The two classic pairs: distance/displacement and speed/velocity

Distance is how far you have travelled in total — the reading on a pedometer, which only ever goes up. It is a scalar. Displacement is the straight-line vector from where you started to where you finished, with its direction. Walk all the way round a 400\ \text{m} running track and stop where you began: your distance is 400\ \text{m}, but your displacement is 0\ \text{m} — you ended up exactly where you set off, so there is no arrow to draw at all.

Worked example — there and back. You walk 3\ \text{km} east, then turn and walk 1\ \text{km} back west.

• Distance travelled: 3 + 1 = 4\ \text{km} (you add every step you took).
• Displacement: 3 - 1 = 2\ \text{km east} (start to finish, in a stated direction).

The same split applies to speed and velocity. Speed is a scalar — just "how fast", a number with a unit. Velocity is a vector — speed with a direction. A plane cruising at 300\ \text{m/s} due north and one cruising at 300\ \text{m/s} due south have the same speed but opposite velocities — and if they're at the same height that difference is the difference between a safe flight and a head-on crash. Direction is not a decoration; it changes what happens.

Adding vectors along a line

When two vectors lie along the same straight line, combining them is easy — you just have to respect the arrowheads. Pick one direction to call positive (say "right is +"), then:

• Vectors pointing the same way? Add their magnitudes; the resultant points that way too.
• Vectors pointing opposite ways? Subtract the smaller from the bigger; the resultant points the way the bigger one pointed.

The single combined vector is called the resultant.

Worked example 1 — a tail-wind. A plane flies at 250\ \text{m/s} east and a wind blows it a further 30\ \text{m/s} east. Same direction, so add:

Worked example 2 — a head-wind. The same plane flies 250\ \text{m/s} east but now into a 30\ \text{m/s} westward wind. Opposite directions, so subtract:

Worked example 3 — a swimmer against the current. A swimmer moves at 2\ \text{m/s} forwards while a river current pushes them 3\ \text{m/s} backwards along the same line. Taking forwards as positive:

a resultant of 1\ \text{m/s} backwards — the current wins, and the swimmer is carried downstream. The minus sign didn't need decoding by hand: it simply announced the direction.

Vectors at right angles: build a triangle

What if two vectors are not on the same line — say one points north and one points east? You can't add or subtract them along a line, because they're neither helping nor fighting directly. Instead you lay them tip-to-tail: draw the first arrow, then start the second arrow from the first one's head. The resultant is the single arrow from the very start to the very end — the slanted third side of the triangle.

When the two vectors are at right angles, that triangle is right-angled, so its longest side (the resultant) is found with Pythagoras:

Worked example — a boat crossing a river. A boat is driven 4\ \text{m/s} straight across a river while the current sweeps it 3\ \text{m/s} downstream, at right angles to its heading. Its resultant velocity over the ground has magnitude

pointing off at an angle between "across" and "downstream" — a tidy 3-4-5 triangle. (You can also find that resultant without any algebra: draw both arrows to scale on graph paper, complete the triangle, and measure the long side with a ruler. That's a scale drawing, and it gives the same 5\ \text{m/s}.) The rule to carry away: same-line vectors add or subtract, but vectors at an angle build a triangle.

Try it: add two vectors tip-to-tail

Set the "across" and "up" parts of two vectors \mathbf{a} (blue) and \mathbf{b} (green) with the sliders. The green arrow is drawn tip-to-tail, starting where the blue one ends, and the bold resultant \mathbf{a}+\mathbf{b} is the arrow straight from the origin to the final tip. Its length — the magnitude — is measured live above.

• Make both vectors point the same way along a line (set their "up" parts to 0): the resultant is just the two lengths added.
• Point them opposite ways along the line: the resultant is the difference.
• Make one purely across and the other purely up: you get a right-angled triangle, and the resultant is the Pythagoras hypotenuse — try (4, 0) and (0, 3) for a 5.