A complete guide to vector notation, column vectors, addition, subtraction, scalar multiplication, magnitude, and vector geometry proofs.
A vector is a quantity with both magnitude (size) and direction — unlike a plain number (a scalar), which only has size. Vectors are used to describe movement, forces, and positions, and they follow their own arithmetic rules for addition, subtraction and scaling. This guide builds vector notation and arithmetic from the ground up, then applies vectors to geometric problems such as finding midpoints and proving that lines are parallel.
A vector represents a movement or displacement: it has a specific length (how far) and a specific direction (which way). Two vectors are considered equal if they have the same magnitude and direction, even if drawn starting at different points on a page.
Vectors are written in several equivalent ways: as a bold lowercase letter (a), an arrow over two capital letters showing start and end points (→AB, from A to B), or as a column vector.
A column vector shows the horizontal and vertical movement separately, written as a pair of numbers stacked vertically, e.g. (3 over 4) means "3 units across, 4 units up." A negative number means movement in the opposite direction (left or down).
To add two column vectors, add their corresponding top and bottom components separately. Geometrically, this is the "nose-to-tail" method: draw the second vector starting where the first one ends; the sum is the single vector from the very start to the very end.
a = (2 over 5), b = (4 over −1). Find a + b.
a + b = (2+4 over 5+(−1)) = (6 over 4).
Subtracting a vector is the same as adding its negative (reversing its direction). To subtract column vectors, subtract the corresponding components.
a = (7 over 2), b = (3 over 6). Find a − b.
a − b = (7−3 over 2−6) = (4 over −4).
Multiplying a vector by a number (a scalar) scales its length without changing its direction (unless the scalar is negative, which reverses the direction). Each component of the column vector is multiplied by the scalar.
a = (3 over −2). Find 4a and −2a.
4a = (12 over −8). −2a = (−6 over 4) — note the reversed direction.
A position vector describes the location of a point relative to a fixed origin, O. The position vector of point A is written →OA, often shortened to just a.
If the position vectors of two points A and B are known (a and b), the vector from A to B is found by subtraction: →AB = b − a. This is one of the most frequently used vector formulas in geometry problems.
A has position vector a = (2 over 3), and B has position vector b = (8 over 1). Find →AB.
→AB = b − a = (8−2 over 1−3) = (6 over −2).
The magnitude (length) of a column vector (x over y) is found using Pythagoras' theorem: |v| = √(x² + y²).
Find the magnitude of v = (5 over 12).
|v| = √(5² + 12²) = √(25 + 144) = √169 = 13.
Two vectors are parallel if one is a scalar multiple of the other (e.g. b = 3a). Parallel vectors point in the same direction (or exactly opposite directions if the scalar is negative), and this property is often used to prove that three points lie on a straight line.
Are u = (4 over 6) and v = (6 over 9) parallel?
v = 1.5u, since 4 × 1.5 = 6 and 6 × 1.5 = 9. Since v is a scalar multiple of u, the vectors are parallel.
The position vector of the midpoint M of a line segment AB is the average of the two endpoint position vectors: m = ½(a + b). More generally, a point dividing AB in a given ratio can be found using a weighted combination of a and b.
A has position vector (2 over 4), and B has position vector (10 over 8). Find the position vector of the midpoint M of AB.
m = ½[(2 over 4) + (10 over 8)] = ½(12 over 12) = (6 over 6).
Vector proofs typically express every relevant line segment in terms of just two starting vectors (often the sides of a triangle), then use vector addition, subtraction and the parallel-vector test to prove a required geometric fact, such as showing two lines are parallel or that a point is the midpoint of a segment.
OABC is a parallelogram with →OA = a and →OC = c. M is the midpoint of AB. Show that →OM is parallel to a diagonal-related direction by expressing it in terms of a and c.
Since OABC is a parallelogram, →CB = →OA = a, so B has position vector c + a. M is the midpoint of AB, so m = ½(a + (c + a)) = ½(2a + c) = a + ½c. This expresses M's position fully in terms of the two starting vectors, ready for further comparison in a full proof.
Common errors include: adding or subtracting column vectors component-wise incorrectly (mixing up the top and bottom rows); forgetting that →AB = b − a, not a − b (the direction matters — it always ends at the second-named point); assuming two vectors are parallel just because they look similar, without checking the scalar multiple relationship precisely; and forgetting to take the square root when calculating magnitude.
| Operation | Rule |
|---|---|
| Addition | Add corresponding components |
| Subtraction | Subtract corresponding components |
| Scalar multiplication | Multiply every component by the scalar |
| Vector from A to B | →AB = b − a |
| Magnitude of (x, y) | √(x² + y²) |
| Midpoint of AB | ½(a + b) |
| Parallel vectors | One is a scalar multiple of the other |
This article is original EDUSAMBAM educational writing. It is designed as a broad vectors resource covering notation, arithmetic and geometric applications for confident problem-solving. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: master column vector addition, subtraction and scalar multiplication first, then practise expressing unknown position vectors in terms of two given vectors before attempting full geometric proofs.
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