Vectors for Beginners: Adding, Subtracting, and Components

Quick answer: A vector has both size and direction (think "3 across, 4 up"), while a scalar has only size (like "5 metres"). You add or subtract column vectors by working on the top numbers and bottom numbers separately, multiply by a scalar by multiplying both numbers, and find a vector's magnitude (length) with sqrt(x^2 + y^2). Get those four moves right and you can handle almost every beginner vectors question.

I'm Rabail, and I sit IGCSE and A-Levels. When I first met vectors my teacher drew a forest of arrows on the board and I nodded along without understanding a thing. What finally made it click was hearing one sentence out loud: a vector is just an instruction for moving - "go this far in this direction." Once I read every column vector that way, the arithmetic stopped feeling abstract. Here's everything I wish someone had put in one place, with the exam-board habits baked in (Edexcel and AQA love the "show that" proof; Cambridge and CBSE lean on position vectors).

Vectors vs scalars

• Scalar: size only. Mass 5 kg, distance 12 m, temperature 30 degrees. Just a number with a unit.
• Vector: size and direction. Velocity 12 m/s due north, a force of 20 N at 30 degrees, a translation "3 right and 4 up."

Quick test: if reversing the direction changes the meaning, it's a vector. Distance is a scalar; displacement (distance in a set direction) is a vector. Speed is a scalar; velocity is a vector. Examiners at every board love that speed-versus-velocity distinction - it's a free mark if you keep it straight.

Column vectors: the notation everything runs on

I'll write a column vector as (x, y): the top number x is the horizontal step (right is positive, left negative), and the bottom number y is the vertical step (up positive, down negative). At A-Level you'll also see (3, 4) written as 3i + 4j - exactly the same thing, where i means one step right and j means one step up.

Adding and subtracting

The rule: tops with tops, bottoms with bottoms. Never mix a top with a bottom.

Adding (5, 2) + (-3, 4): top is 5 + (-3) = 2, bottom is 2 + 4 = 6, so the answer is (2, 6).

Subtraction is where order matters. The vector from A to B is written AB and equals (position of B) - (position of A) - end minus start. Get that order backwards and your arrow points the wrong way, which quietly wrecks every proof built on top of it. If A = (1, 3) and B = (4, 1): AB = (4-1, 1-3) = (3, -2). And BA = (-3, 2), which is just -AB.

Scalar multiples

Multiply both numbers by the scalar. So 3 times (2, -1) = (6, -3). The direction doesn't change; the arrow just gets three times longer. A negative scalar flips it: -2 times (2, -1) = (-4, 2) - same line, opposite way, twice the length. This one idea sits behind every "prove these are parallel" question: two vectors are parallel exactly when one is a scalar multiple of the other.

Magnitude (the length of a vector)

The magnitude is just Pythagoras on the two components: the magnitude of (x, y) equals sqrt(x^2 + y^2). It's always positive.

Worked: the magnitude of (6, -8) = sqrt(6^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10. Notice the minus vanishes the moment you square it - a classic slip is dragging it through and getting sqrt(36 - 64). If a magnitude ever comes out negative or as the root of a negative, you know you've made that mistake. If a particular step confuses you, our free /explain tool will break it down line by line.

Resolving into components (AP Physics and A-Level)

When you're given a size plus an angle, you split the vector into its horizontal and vertical parts. Take a force of 20 N acting at 30 degrees above the horizontal:

1. Horizontal part = 20 cos(30) = 17.3 N (round to the question's accuracy).
2. Vertical part = 20 sin(30) = 10 N.

The pattern: cos goes with the part next to the angle (here, horizontal), sin with the part facing it (vertical). If the angle were measured from the vertical instead, you'd swap them. AP Physics 1 and A-Level Mechanics both run on this move - every inclined-plane and projectile question opens by resolving. Feed a resolve-this problem into our free /math-solver and it'll show each step rather than just the answer.

The "show that" geometry proof

This is the question Edexcel and AQA park at the end of the higher paper for the final few marks, and Cambridge IGCSE loves it too. The classic version: prove that three points lie on one straight line (collinear).

Worked example - show that A(1, 2), B(3, 5) and C(7, 11) are collinear:

1. Find AB: B - A = (3-1, 5-2) = (2, 3).
2. Find BC: C - B = (7-3, 11-5) = (4, 6).
3. Spot the scalar multiple: (4, 6) = 2 times (2, 3), so BC = 2 times AB. That makes AB and BC parallel.
4. Conclude in words: AB and BC are parallel and they share the point B, so A, B and C lie on a single straight line.

Step 4 is where the marks actually live. Almost everyone finds the scalar multiple; the students who drop the mark forget to say "and they share a common point." Parallel on its own only proves parallel - you need the shared point to prove collinear. Write that sentence every time.

Test yourself

1. Work out (7, -1) - (2, 4).
2. Find the magnitude of (-5, 12).
3. P = (2, 1), Q = (4, 5), R = (7, 11). Are P, Q and R collinear?

Answers:

1. (5, -5).
2. sqrt((-5)^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13.
3. PQ = (2, 4) and QR = (3, 6). Since (3, 6) = 1.5 times (2, 4), QR = 1.5 times PQ, so they're parallel and share Q - yes, collinear.

Want these marked and explained rather than just checked? Drop any of them into the free /math-solver and it'll show every line; then turn whatever tripped you up into a quick set with /flashcards.

FAQ

Is a column vector the same as coordinates?

They look identical but mean different things. Coordinates (3, 4) are a fixed spot on the grid. A column vector (3, 4) is a movement - 3 right and 4 up - from wherever you happen to start. A position vector is the special case that runs from the origin to a point, so it just happens to match that point's coordinates.

How do I know whether to use sin or cos when resolving?

Line the angle up with the side you want. The component next to the angle (adjacent) uses cos; the component facing the angle (opposite) uses sin. If the angle is measured from the vertical rather than the horizontal, the two simply swap.

What is a unit vector?

A vector with magnitude 1. To build one, divide a vector by its own magnitude: (6, 8) has magnitude 10, so its unit vector is (0.6, 0.8). Cambridge A-Level and CBSE ask for these directly, and they're how you describe pure direction.

Do CBSE and A-Level go further than this?

Yes - the dot (scalar) product, the cross product, and 3D vectors written (x, y, z). But every one of those is built on the same four moves: adding, subtracting, scalar multiples and magnitude. Get these solid first and the harder topics have somewhere to stand.

In short: Read a vector as an instruction to move a set distance in a set direction. Add and subtract componentwise (end minus start, in that order), scale both numbers together, and use sqrt(x^2 + y^2) for length. Master those four moves and the "show that" proofs are just the same moves with one concluding sentence tacked on the end.