Vectors
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Vectors
A vector is just an instruction for how far to move and in which direction — and once you can add, subtract, scale, and measure the length of those instructions, this whole chapter is yours.
Revision Guide · Column Vectors · Vector Diagrams · Magnitude
Summary — What This Chapter Covers
- Column vectors describe a movement from one point to another — a top number for left/right, a bottom number for up/down.
- You can add and subtract column vectors by combining the top numbers together and the bottom numbers together.
- You can multiply a vector by a scalar (an ordinary number) — this stretches, shrinks, or flips the vector.
- Vectors can be drawn as arrows on a diagram — the length shows magnitude, the arrowhead shows direction.
- A vector from point to point is written
AB→, and it points the way toBA→— same length, reversed direction. - The magnitude (length) of a vector is found using Pythagoras' theorem on its two components.
- Multiplying by a negative scalar reverses the direction of a vector as well as changing its length.
1. Introduction to Column Vectors
Imagine you're giving someone directions across a grid — not "turn left at the shop," but pure movement: That's exactly what a column vector is: a set of travel instructions written as a little vertical pair of numbers.
Column vectors are also called translation vectors, because they describe exactly the same thing as a translation in geometry — a slide, with no rotation and no resizing. If you've done translations before, you already half-know this topic.
This is the easiest operation in the whole chapter, because there's no trick — you just deal with the top numbers together, and the bottom numbers together, completely separately. Think of it like two separate number lines running in parallel: one for "sideways total," one for "up/down total."
For example:
2 − (−1) is 2 + 1 = 3, not 1. Always rewrite double negatives before calculating.
A scalar is just an ordinary number — it has size but no direction (that's the whole point of the name: it "scales" things up or down, it doesn't point anywhere). When you multiply a vector by a scalar, you multiply components by that number.
Where this gets slightly trickier — and where exam questions like to test you — is when an expression mixes scalar multiplication addition/subtraction in one line. The fix is simply order of operations: multiply first, then add or subtract.
Step 2: Add or subtract the resulting column vectors (top with top, bottom with bottom).
Step 1 — write as one vector: 2a = (2p, 6) and 3b = (−6, 3), so 2a + 3b = (2p − 6, 9).
Step 2 — this equals (4, q): so (2p − 6, 9) = (4, q).
Step 3 — match components: top numbers: 2p − 6 = 4 → 2p = 10 → p = 5. Bottom numbers: 9 = q → q = 9.
This "match the top with the top, and the bottom with the bottom" idea — two equal vectors must have equal components — is one of the most useful exam tricks in this whole topic.
2. Representing Vectors as Diagrams
A vector has two properties: a magnitude (its size — how far it goes) and a direction (which way it's pointing). To show both on a diagram, you draw a straight line whose represents the magnitude, and an at the end to show which way it's pointing. Without the arrow, it's just a line segment — it's the arrow that turns it into a vector.
If a vector starts at point and ends at point , we write it as AB→ (an arrow drawn over "AB", pointing right). The order of the letters matters enormously here:
You can draw a vector on a grid — its position doesn't matter, only its length and direction do (this is a really important idea: two vectors with the same components represent the "same" vector even if drawn in totally different places on the page).
2. Count 3 units right and 4 units up from that dot, and mark a second dot.
3. Draw a straight line connecting the two dots.
4. Add an arrowhead pointing toward the dot (the one you counted to).
Watch out for negative numbers and zeros — they're where students most often draw the wrong picture:
You already know the arithmetic rule (multiply both components). Here's what it actually like on a diagram, which is where a lot of understanding really clicks into place.
This is often called the nose-to-tail method, and it's the visual version of vector addition. Picture walking along one vector, then immediately continuing your walk along the second vector from wherever the first one left you. The "total journey" — from your very first starting point to your very last ending point — is the vector a + b.
2. Starting exactly where a ended (nose-to-tail — the "nose"/arrowhead of a meets the "tail" of b), draw vector b.
3. Draw a new line straight from the very start of a to the very end of b. That line, with an arrowhead at the end, IS the vector a + b.
Subtraction works the same way, but remember that a − b is really just a + (−b) — so you draw a as normal, then draw b reversed (i.e. −b, pointing the opposite way) starting from the end of a.
2. Starting where a ended, draw −b (same length as b, but pointing the opposite way).
3. Draw a new line from the start of a to the end of −b. That's a − b.
The rule for any vector from point P(x₁, y₁) to point Q(x₂, y₂) is PQ→ = (x₂ − x₁, y₂ − y₁) — end point minus start point, top with top, bottom with bottom.
AB→ = (2 − (−4), 4 − 2) = (6, 2) — 6 right and 2 up.
AC→ = (3 − (−4), −4 − 2) = (7, −6) — 7 right and 6 down.
CB→ = (2 − 3, 4 − (−4)) = (−1, 8) — 1 left and 8 up.
Bonus insight: notice that AB→ + BC→ + CA→ always equals (0, 0) for any three points — because it describes a journey that starts at A, wanders to B, then C, then returns all the way back to A. Since you end up exactly where you started, the total displacement is nothing at all — a closed loop always sums to the zero vector. This is a classic "explain why" exam question, and no calculation is needed to answer it — just describe the journey!
3. Magnitude of a Vector
The magnitude of a vector is simply its — how far you'd actually travel if you walked along it, measured as a straight-line distance. It's also called the modulus of the vector (same idea, fancier name — you'll see both used interchangeably in exams). Two crucial things to remember about magnitude:
- It's always positive (or zero) — length can't be negative, even if the vector's components are negative.
- It ignores direction completely — the magnitude only cares about "how far," not "which way." That's why |AB→| = |BA→| — reversing a vector doesn't change how long it is.
Why does this work? Picture the vector as the diagonal (hypotenuse) of a right-angled triangle: the horizontal component (x) is one shorter side, the vertical component (y) is the other shorter side, and the vector itself is the hypotenuse connecting them. Pythagoras' theorem — the same one you use for right-angled triangles everywhere else in maths — gives you the length of that hypotenuse directly from the two shorter sides.
(a) Find AB→ as a column vector.
AB→ = (end − start) = (7 − (−3), 1 − 5) = (10, −4).
(b) Find the magnitude of AB→.
|AB→| = √(10² + (−4)²) = √(100 + 16) = √116 = 2√29 (leave in surd form unless told to give a decimal).
(c) Briefly explain why |BA→| = |AB→|.
Magnitude measures a vector's only — direction is irrelevant to length. AB→ and BA→ cover exactly the same physical distance between the same two points, just described in opposite directions, so their magnitudes must be identical.
(d) Vector CD→ has three times the magnitude of AB→. Write down a possible column vector for CD→.
Multiplying AB→'s components by 3 (or −3) multiplies its length by exactly 3, while keeping (or reversing) its direction: 3 × (10, −4) = (30, −12), or equally valid, −3 × (10, −4) = (−30, 12). Either answer is correct because the question only fixed the magnitude, not the direction.
What to Memorise
| Term / Formula | What it means |
|---|---|
| ( x, y ) | Column vector: x = horizontal movement (+ right, − left), y = vertical movement (+ up, − down) |
| (a,b) + (c,d) | = (a+c, b+d) — add tops together, add bottoms together |
| (a,b) − (c,d) | = (a−c, b−d) — subtract tops together, subtract bottoms together |
| k(x, y) | = (kx, ky) — scalar multiplication: multiply both components by k |
| Scalar | An ordinary number with size but no direction (used to scale a vector) |
| AB→ | Vector starting at A, ending at B. AB→ = (Bx − Ax, By − Ay) |
| BA→ = −AB→ | Same length as AB→, but points the exact opposite direction |
| Magnitude / Modulus | The length of a vector — always positive, direction-independent |
| |a| = √(x² + y²) | Pythagoras' theorem applied to the vector's components |
| Nose-to-tail | Method for drawing a + b: draw a, then draw b starting where a ended |
| Negative scalar | Reverses vector direction AND scales the length by the number's size |
| Closed loop rule | AB→ + BC→ + CA→ = (0,0) — any journey that returns to its start sums to zero |
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