HookWhy Channel swimmers never swim in a straight line
The shortest crossing of the Dover Strait, from Shakespeare Beach near Dover to Cap Gris Nez in France, is only about 21 miles (34 km) in a dead straight line — and yet nobody who has ever swum the Channel has swum 21 miles. Captain Matthew Webb, the first to do it unaided in August 1875, was in the water for nearly 22 hours and traced a long S-shaped path, and modern swimmers still routinely cover several miles more than the straight-line distance. The reason is not that they cannot swim straight. It is that the tide in the Strait runs sideways, first one way for roughly six hours and then the other, and the swimmer's true motion across the sea floor is the combination of where they are swimming and where the water is carrying them.
That combination has a name in mathematics: a vector, and adding two of them together gives the resultant — the swimmer's actual path. A vector is any quantity with both a size and a direction: a displacement of '3 metres east', a velocity, a push. This section teaches you to write vectors as columns of numbers, to use them to describe translations, to add and subtract them, and to multiply them by a plain number to scale them. It is one of the shorter topics on the Foundation papers, and one of the most reliable for marks, because the arithmetic is small and the rules are few — as long as you never lose track of a sign.
ModelWhat a vector is — and the column that describes it
A vector carries two pieces of information at once: how far, and in which direction. That is different from an ordinary number (a scalar, like a temperature or a mass), which has size only. The cleanest way to write a 2D vector is as a column vector: two numbers stacked in a bracket, the top number telling you how far to move right (or left, if negative), and the bottom number how far to move up (or down, if negative). So the column vector with 4 on top and 3 below means 'go 4 right and 3 up'.
A translation — the transformation that slides a shape without turning or resizing it — is described by exactly this kind of vector. To translate a shape 'by the vector 4 right, 3 down' you move every single point of it 4 across and 3 down, so the shape lands in a new position, identical in size and orientation. This is the first place column vectors appear on the paper, and the description must be complete: the word 'translation' AND the vector. Naming the vector but calling the move a 'shift', or describing a slide as though it were a rotation, throws the marks away.
A triangle has a vertex at the point (1, 2). After a translation it lands at (5, −1). What is the translation vector? Compare the coordinates: the x went from 1 to 5, a change of +4; the y went from 2 to −1, a change of −3. So the translation vector is 4 on top and −3 below — 4 right and 3 down. To reverse the translation you simply flip both signs: −4 and +3 sends the shape back where it started.
MechanismAdding and subtracting — do it component by component
Adding two vectors means doing one journey and then the other, and the arithmetic could not be simpler: add the tops together and add the bottoms together. If you go 3 right and 1 up, then 2 left and 4 up, your overall journey is 1 right and 5 up — the resultant. That is the Channel-swim idea made exact: the swimmer's stroke is one vector, the tide is another, and their sum is the path over the ground.
Subtracting works the same way, component by component: subtract the tops, subtract the bottoms. Watch the signs like a hawk, because subtracting a negative flips it to a plus — this is where almost every lost mark in the topic comes from. Subtracting the vector (3, −1) means taking away 3 from the top and taking away −1 from the bottom, and taking away −1 is the same as adding 1.
There is a neat geometric picture behind subtraction: the vector from point A to point B, written as the journey you make to get from one to the other, is found by 'destination minus start'. Reverse the direction and every sign flips — the trip from B back to A is the exact negative of the trip from A to B.
Add the vectors (3, 1) and (−2, 4). Tops: 3 + (−2) = 1. Bottoms: 1 + 4 = 5. Resultant: (1, 5) — one right, five up. Now subtract: (5, 2) − (3, −1). Tops: 5 − 3 = 2. Bottoms: 2 − (−1) = 2 + 1 = 3. Answer: (2, 3). The bottom is the trap — subtracting −1 adds 1, so it becomes 3, not 1. Write the intermediate step 2 − (−1) = 2 + 1 out in full so the sign change is on the page.
ModelScalar multiplication — stretching a vector
Multiplying a vector by an ordinary number (a scalar) stretches or shrinks it, and multiplies both components. Doubling the vector (2, −1) gives (4, −2): the top and the bottom are each multiplied by 2. The commonest error is to double only one of the two numbers — but a scalar acts on the whole vector, top and bottom together.
Geometrically, multiplying by a positive number keeps the direction exactly the same and only changes the length: 3 times a vector points the same way but is three times as long, and ½ times a vector points the same way but is half as long. Multiplying by a negative number reverses the direction and scales the length — so −1 times a vector is the same arrow pointing the opposite way, which is precisely the 'B to A is the negative of A to B' fact from before.
Two vectors that are scalar multiples of each other are parallel, because they point along the same line (or the exact opposite direction). That is how you can tell at a glance that (2, 3) and (4, 6) are parallel — the second is just 2 times the first.
Work out 3 times the vector (2, −1), then combine: find a + 2b where a = (1, 3) and b = (2, −1). First 3 × (2, −1): multiply both parts, giving (6, −3). Now 2b = 2 × (2, −1) = (4, −2), and adding a: (1, 3) + (4, −2) = (1 + 4, 3 + (−2)) = (5, 1). Do the scalar multiplication first, then the addition — the same order-of-operations discipline you use with numbers.
CaseThe diagram — nose to tail, and back to the tide
Vectors can be drawn as well as calculated, and the picture makes the rules obvious. A vector is drawn as an arrow: its length shows the size, its direction shows the direction. To add two vectors on a diagram you place them nose to tail — draw the first arrow, then start the second where the first ended — and the resultant is the single arrow from the very start to the very end. This is why the two-stage journey (3 right 1 up, then 2 left 4 up) lands you at the resultant (1 right, 5 up).
Reversing a vector flips its arrow to point the other way, which is what the negative sign does. Subtracting b from a can be pictured as adding the reversed arrow of b onto a.
The Channel swimmer is exactly this diagram in salt water. Draw the swimmer's own effort as one arrow pointing at France, draw the tide as a second arrow pointing along the coast, place them nose to tail, and the resultant arrow points where the swimmer actually ends up — downstream of the target. Aim straight and the tide sweeps you sideways; the S-shaped path is the swimmer constantly re-aiming to fight a resultant that keeps changing as the tide turns.
MechanismJourneys on a grid — combining several vectors
Real questions often chain three or four vectors together — a walk from a car park to a bench to a gate to a pond — and ask for the single resultant vector of the whole trip, or for the return journey. The method scales up cleanly: add all the tops, add all the bottoms, once. Because addition does not care about order, you can total every 'right/left' figure and every 'up/down' figure separately.
A useful check drops out for free: if a journey ends where it began — a full loop back to the start — the resultant must be the zero vector, 0 on top and 0 below, because you have gone nowhere overall. So the outward vectors and the return vectors of any closed loop must cancel exactly, and that is a quick way to test whether you have added correctly.
A robot moves by (4, 2), then (−1, 3), then (−2, −1). Find the single vector for the whole journey. Add the tops: 4 + (−1) + (−2) = 1. Add the bottoms: 2 + 3 + (−1) = 4. The resultant is (1, 4). To send the robot straight back to its start, apply the negative: (−1, −4). Check: (1, 4) + (−1, −4) = (0, 0), the zero vector — it has returned to the start, exactly as a closed loop should.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
Vectors are a small, dependable source of marks on the Foundation papers, and they appear on both the non-calculator and calculator papers because the arithmetic is deliberately light — the skill being tested is care, not computation. Do every addition and subtraction component by component and write the intermediate line out, especially where a sign changes: showing 2 − (−1) = 2 + 1 = 3 on the page earns the method mark even if you fumble the final total.
When the question is 'describe fully the single transformation', a translation must be named as a translation and given its column vector — both halves are needed, and calling it a rotation or reflection scores zero even with the correct vector. When you are handed a translation and asked to reverse it, or to find the vector between two points, remember 'destination minus start' and flip all the signs for the return trip.
Watch the two classic traps the mark scheme is built around: multiplying a vector by a scalar acts on both numbers, not one; and subtracting a negative adds. At Foundation you will not be asked to prove points are in a straight line or to argue geometrically with vectors — those vector proofs are Higher-tier — so if a question seems to want an algebraic proof, you are almost certainly reading a Higher paper by mistake. Keep the work tidy, keep the columns aligned, and the signs will look after themselves.