A column vector (x over y) describes a movement, not a position: x is how far right (or left, if negative) and y is how far up (or down). Between two coordinates A and B, the vector AB is always B minus A, componentwise. Its magnitude, written |AB|, is √(x² + y²), which is just Pythagoras applied to the horizontal and vertical parts of the movement.
A coordinate like (3, 5) tells you a fixed position on the grid. A column vector that looks identical, (3 over 5), tells you something completely different: a displacement of 3 units across and 5 units up, from wherever you start. The two use similar-looking bracket notation, so students often treat a vector as if it were a point, or forget that the same vector can start from anywhere and still mean the same movement.
This mix-up is exactly what causes the most common exam error: computing vector AB as A − B instead of B − A. Getting the order backwards flips the direction of the vector and every answer built on it, including magnitude problems that ask for a length (magnitude has no negative sign, so a direction error can slip through unnoticed until the final answer looks odd). Column vectors are a separate topic from matrices, though both use similar bracket notation, so keep the two skills distinct in your revision.
Step 1
Write down coordinates A and B
Step 2
Vector AB = B − A, componentwise
Step 3
Write the result as (x over y)
Step 4
Magnitude = √(x² + y²)
Addition and subtraction of column vectors work the same way, componentwise: add or subtract the top numbers together and the bottom numbers together. A scalar multiple like 3v stretches the vector to three times its length in the same direction, and −v reverses it, same length, opposite direction. None of this changes how you handle the components: work on x and y separately, then combine.
A is the point (2, 3) and B is the point (6, 9). Find the column vector AB and its magnitude.
Solution
AB = (4 over 6)|AB| = √(4² + 6²) = √(16 + 36) = √52.√52 = √4 × √13 = 2√13.Quick check: if you had computed A − B instead, you would get (−4 over −6), same magnitude but pointing the wrong way. Always ask yourself which letter comes second in the vector's name, that is your finishing point, and subtract the starting point from it.
Vector a = (x over 3) and vector b = (1 over y). Given that 2a − b = (7 over 1), find x and y.
Solution
2a first, a scalar multiple doubles every component: 2a = (2x over 6).b componentwise: 2a − b = (2x − 1 over 6 − y).(7 over 1), giving two separate equations: 2x − 1 = 7 and 6 − y = 1.2x − 1 = 7, 2x = 8, so x = 4. From 6 − y = 1, y = 5.The step students get wrong
Two mistakes account for most lost marks here. First, computing AB as A − B instead of B − A, which reverses the direction and (in longer questions) reverses every downstream answer. Second, sign slips when squaring negative components inside a magnitude: (−6)² is 36, not −36, because squaring a negative number always gives a positive result. If a component is negative, square it in brackets on your working paper before you add the two squares together, so the negative sign cannot get lost.
You can verify √(x² + y²) quickly on the calculator instead of doing the squaring by hand, which is useful for checking your working under time pressure.
On the fx-97SG CW and X
Is a column vector the same as a coordinate?
No. A coordinate is a fixed position on the grid. A column vector is a movement: how far across and how far up or down. The same vector can start from any point on the grid and still describe the same movement, which is why two parallel lines of the same length can share the same column vector even though their coordinates are completely different.
Why is vector AB equal to B minus A and not A minus B?
Vector AB means the movement that takes you from A to B. To get from A's coordinates to B's coordinates, you add the vector to A, so the vector itself must be B minus A. If you calculate A minus B instead, you get the vector BA, which points the opposite way.
Does the magnitude of a vector depend on where it starts?
No. Magnitude only depends on the components (the x and y values of the column vector), not on the coordinates of the start or end point. Two vectors with the same components have the same magnitude and the same direction, even if they are drawn starting from completely different points on the grid.
How is this connected to the distance between two points?
They use the same idea. The magnitude of vector AB is exactly the distance from A to B, so finding |AB| and finding the distance between two coordinates are the same calculation done two different ways. See our guide on the distance and midpoint formula for the coordinate-geometry version of this. Once magnitude feels comfortable, the natural next step is using vectors to prove three points are collinear.
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