1The idea in one paragraph

Some quantities are fixed by a single number: a mass of 4 kg, a temperature of 20 °C. They are scalars. Others need a size and a direction: a displacement of 5 km north-east, a force of 30 N straight down. They are vectors. A vector does not care where it starts, only how far and which way, so it is drawn as an arrow that can be slid anywhere without changing it. Written in components, a vector lists how far it goes along each axis: 3i + 2j means 3 along x and 2 along y. Adding vectors means following one arrow and then the next, which is the same as adding their components. Multiplying by a number stretches the arrow and, if the number is negative, turns it round. Two arrows that are multiples of each other are parallel. The length of a vector comes from Pythagoras. And because AB⃗ = b − a, every question about points turns into arithmetic on their position vectors, which is how vectors prove facts about triangles and parallelograms.

2What a vector is

A vector is a quantity with magnitude (size) and direction. A scalar has magnitude only. Displacement, velocity, force and acceleration are vectors; distance, speed, mass and time are scalars.

A vector is drawn as a directed line segment: an arrow whose length shows the magnitude and whose arrowhead shows the direction. The arrow from A to B is AB⃗. A is its tail and B its head. Figure 1 shows the key idea: three arrows in different places are the same vector if they have the same length and the same direction. Position on the page is not part of a vector.

Two consequences follow at once.

• The negative of a vector, −v, has the same length as v and the opposite direction. In particular BA⃗ = −AB⃗.
• The zero vector, 0, has magnitude 0 and no direction. It is what you get from AA⃗, or from v + (−v).

3Components and base vectors

On a coordinate grid, a vector is described by how far it moves in each direction. The base vectors are unit arrows along the axes: i = one step in the x-direction, j = one step in the y-direction and, in three dimensions, k = one step in the z-direction.

Figure 2 draws v = 3i + 2j: three steps of i, then two steps of j. The numbers 3 and 2 are the components of v, and as a column they are written one above the other.

In three dimensions a vector has three components. v = v₁i + v₂j + v₃k, with v₁, v₂ and v₃ in a column, means v₁ along x, v₂ along y and v₃ along z. The axes are drawn right-handed: if x points towards you and y to the right, z points up, as in Figure 3.

Position vectors. The position vector of a point P is the vector OP⃗ from the origin O to P. Its components are the coordinates of P. So the point P(2, 3, 4) has position vector p = OP⃗ = 2i + 3j + 4k. The letters follow a convention: the position vector of A is a, of B is b, and so on. A point and its position vector carry the same three numbers, but they are different things: P is a place, p is the journey from O to that place. A missing component is zero: 5i − k has components 5, 0 and −1.

4Adding and subtracting

Geometrically, to add u and v, draw v starting where u ends. The sum u + v goes from the tail of u to the head of v. This is the triangle law, Figure 4(a). Drawing both from the same point and completing the parallelogram gives the same diagonal, Figure 4(b), and shows that u + v = v + u.

Algebraically, add the components. A journey of 3 along x followed by 1 along x is 4 along x, whatever else happens.

Subtracting is adding the negative: u − v = u + (−v). Geometrically, u − v is the arrow from the head of v to the head of u when both start at the same point.

The displacement from A to B. Figure 5 gives the single most used fact on this page. To go from A to B, go back from A to O (that is −a), then out from O to B (that is b).

AB⃗ = b − a · the position of the end minus the position of the start.

Worked example 1 (Paper 1). A(2, −1, 3) and B(5, 3, −9). Find AB⃗ and BA⃗.

Getting the order right is the whole question: end minus start. a − b gives BA⃗, the same length pointing the wrong way.

5Scalar multiples and parallel vectors

Multiplying a vector by a scalar k multiplies every component by k: kv = kv₁i + kv₂j + kv₃k. Geometrically, kv is v stretched by the factor |k|, and turned round if k < 0. Figure 6 shows v, 2v, ½v and −1.5v.

That gives the test for direction:

u and v are parallel exactly when u = kv for some scalar k ≠ 0. Same direction if k > 0, opposite if k < 0.

Worked example 2 (Paper 1). Show that u = 2i − 3j + k and w = −6i + 9j − 3k are parallel. Then find p and q so that 4i + pj + qk is parallel to u.

Always find the scalar from one component and check it in all the others. If the components do not share one ratio, the vectors are not parallel.

Collinear points. Three points A, B and C lie on one straight line (are collinear) when AB⃗ and AC⃗ are parallel. They share the point A, so parallel means the same line, not two parallel lines.

Worked example 3 (Paper 1). Show that A(1, 2, 3), B(3, 5, 4) and C(7, 11, 6) are collinear, and find the ratio AB : BC.

The written conclusion earns the reasoning mark: "AC⃗ = 3AB⃗, so AB and AC are parallel, and since they share the point A, the points are collinear."

6Magnitude, distance and unit vectors

The magnitude |v| is the length of the arrow. It comes from Pythagoras, applied once in two dimensions and twice in three.

|v| = √(v₁² + v₂² + v₃²) · in the formula booklet. The distance between A and B is |AB⃗|.

Worked example 4 (Paper 1). Find the distance between A(2, −1, 3) and B(5, 3, −9).

A unit vector has magnitude 1. To make a unit vector in the direction of v, divide v by its own length: the unit vector is v ÷ |v|. To get a vector of any length L in that direction, multiply the unit vector by L.

Worked example 5 (Paper 1). v = 2i − j + 2k. Find a unit vector in the direction of v, and a vector of magnitude 12 in the direction opposite to v.

An unknown component can be found from a magnitude. If |ti + 2j − k| = 3, then t² + 4 + 1 = 9, so t² = 4 and t = ±2. Both values are valid unless the question restricts t.

7Midpoints and parallelograms

The midpoint. The midpoint M of AB is halfway along AB⃗, so m = a + ½(b − a) = ½(a + b). The same reasoning finds any point along a segment: the point that divides AB in the ratio 1 : 2 is a third of the way along, at a + ⅓(b − a).

The parallelogram. ABCD is a parallelogram (letters in order round the shape) exactly when AB⃗ = DC⃗: opposite sides are equal and parallel, and point the same way when read in that order.

Worked example 6 (Paper 1). A(1, 0, 2), B(4, 1, 0) and C(6, 5, 3) are three vertices of the parallelogram ABCD. Find the coordinates of D.

Writing AB⃗ = CD⃗ instead is the classic slip: it gives the wrong fourth vertex, because CD⃗ points the opposite way to AB⃗ in a parallelogram ABCD.

8Proving geometry with vectors

The guide asks you to prove geometric properties with vectors. The method is always the same: name two independent vectors, write everything else in terms of them, and read off the conclusion. "Parallel" means one vector is a multiple of the other; "equal length" means equal magnitudes; "bisect" means two routes reach the same point.

Worked example 7 (Paper 1). In triangle OAB, M is the midpoint of OA and N the midpoint of OB. Prove that MN is parallel to AB and half its length. This is the midpoint theorem, drawn in Figure 7.

So MN⃗ is a positive multiple of AB⃗: MN is parallel to AB, and |MN⃗| = ½|AB⃗|, so it is half as long. The proof works for every triangle, because a and b were never given values.

Worked example 8 (Paper 1). Prove that the diagonals of a parallelogram bisect each other.

The two midpoints have the same position vector, so they are the same point: each diagonal passes through the midpoint of the other. That is what "bisect each other" means.