Vectors

Vectors describe movement with both magnitude (size) and direction. For Edexcel GCSE Mathematics, all students need to understand column vectors for translations, and Higher tier students must also perform vector geometry proofs — showing that points are collinear, finding midpoints, and determining ratios.

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Key Vocabulary

Term	Meaning
Scalar	A quantity with magnitude only (e.g. speed, mass)
Vector	A quantity with both magnitude and direction (e.g. velocity, displacement)
Column vector	A vector written as (x over y) showing horizontal and vertical components
Magnitude	The length (size) of a vector
Parallel vectors	Vectors that are scalar multiples of each other (same or opposite direction)
Collinear	Points that lie on the same straight line
Position vector	The vector from the origin to a point
Resultant	The single vector that has the same effect as two or more vectors combined

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Column Vectors

A column vector (a over b) means:

• Move a units horizontally (positive = right, negative = left)
• Move b units vertically (positive = up, negative = down)

Worked Example 1

Write the column vector for a movement of 3 right and 5 up.

Vector = (3 over 5) — often written as a column: the top component is 3, the bottom is 5.

Worked Example 2

Write the column vector from A(1, 4) to B(5, 2).

Vector AB = (5 - 1 over 2 - 4) = (4 over -2)

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Adding and Subtracting Vectors

Addition: Add corresponding components.

(a over b) + (c over d) = (a + c over b + d)

Subtraction: Subtract corresponding components.

(a over b) - (c over d) = (a - c over b - d)

Worked Example 3

If a = (3 over 2) and b = (1 over -5), find a + b and a - b.

a + b = (3 + 1 over 2 + (-5)) = (4 over -3) a - b = (3 - 1 over 2 - (-5)) = (2 over 7)

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Scalar Multiplication

To multiply a vector by a scalar, multiply each component.

k x (a over b) = (ka over kb)

Worked Example 4

If a = (2 over -3), find 4a and -2a.

4a = (8 over -12) -2a = (-4 over 6)

Key Fact: If b = ka for some scalar k, then a and b are parallel. If k > 0, they point in the same direction; if k < 0, opposite directions.

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Magnitude of a Vector

The magnitude (length) of vector (a over b) is:

|v| = $\sqrt{a^{2} + b^{2}}$a2+b2​

Worked Example 5

Find the magnitude of (3 over 4).

|v| = $\sqrt{3^{2} + 4^{2}}$32+42​ = $\sqrt{9 + 16}$9+16​ = $\sqrt{25}$25​ = 5

Worked Example 6

Find the magnitude of (-5 over 12).

|v| = $\sqrt{(-5)^{2} + 12^{2}}$(−5)2+122​ = $\sqrt{25 + 144}$25+144​ = $\sqrt{169}$169​ = 13

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Vector Notation

Vectors can be written as:

• A bold letter: a, b (in print)
• A letter with a squiggle underneath (in handwriting): a, b
• Two letters with an arrow: vector AB (from A to B)
• A column vector: (3 over -2)

Important: vector AB means "from A to B." vector BA = -vector AB (same magnitude, opposite direction).

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Vector Pathways

To travel from one point to another, follow a path along known vectors.

Worked Example 7

In triangle OAB, vector OA = a and vector OB = b. M is the midpoint of AB. Find vector OM.

Vector AB = vector AO + vector OB = -a + b = b - a

M is the midpoint of AB, so vector AM = 1/2 x vector AB = 1/2(b - a)

Vector OM = vector OA + vector AM = a + 1/2(b - a) = a + 1/2b - 1/2a = 1/2a + 1/2b = 1/2(a + b)

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Vector Geometry Proofs [H]

Higher tier questions ask you to use vectors to prove geometric results.

Showing Vectors are Parallel

Two vectors are parallel if one is a scalar multiple of the other.

Worked Example 8 [H]

In triangle OAB, vector OA = a and vector OB = b. P is the midpoint of OA and Q is the midpoint of OB. Show that PQ is parallel to AB.

Vector PQ = vector PO + vector OQ = -1/2a + 1/2b = 1/2(b - a) Vector AB = b - a

PQ = 1/2 x AB, so PQ is parallel to AB (and half its length). This is the midpoint theorem.

Showing Points are Collinear

Three points X, Y, Z are collinear if vector XY is a scalar multiple of vector XZ (or vector YZ).

Worked Example 9 [H]

O, A, B are such that vector OA = a and vector OB = b. P is the point such that vector OP = 2a + 3b. Q is the point such that vector OQ = 4a + 6b. Show that O, P and Q are collinear.

Vector OP = 2a + 3b Vector OQ = 4a + 6b = 2(2a + 3b) = 2 x vector OP

Since OQ = 2 x OP, the vectors are parallel and share the point O, so O, P and Q are collinear. Also, OQ = 2 x OP, so P is the midpoint of OQ.

Finding Ratios

Worked Example 10 [H]

In triangle OAB, vector OA = a and vector OB = b. P lies on AB such that AP : PB = 2 : 3. Find vector OP in terms of a and b.

Vector AB = b - a Vector AP = (2/5) x vector AB = (2/5)(b - a) Vector OP = vector OA + vector AP = a + (2/5)(b - a) = a + 2/5b - 2/5a = 3/5a + 2/5b = (1/5)(3a + 2b)

Worked Example 11: Showing Points are Collinear [H]

OABC is a parallelogram where vector OA = a and vector OC = c. P is the midpoint of AB. Q is the point on OB such that OQ : QB = 2 : 1.

Since OABC is a parallelogram, vector OB = a + c and vector AB = c.

Vector OP = vector OA + (1/2) vector AB = a + (1/2)c

Vector OQ = (2/3) vector OB = (2/3)(a + c) = (2/3)a + (2/3)c

To test whether C, P and Q are collinear, find vectors CP and CQ:

Vector CP = vector OP − vector OC = a + (1/2)c − c = a − (1/2)c

Vector CQ = vector OQ − vector OC = (2/3)a + (2/3)c − c = (2/3)a − (1/3)c = (2/3)(a − (1/2)c) = (2/3) × vector CP

Since CQ = (2/3) CP, the vectors are scalar multiples and share the common point C, so C, P and Q are collinear. ∎

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Common Mistakes and Misconceptions

• Getting the direction of a vector wrong. vector AB goes from A to B. If you want to go from B to A, negate it: vector BA = -vector AB.
• Forgetting to negate when going "against" a vector. In pathways, if you travel from A to O and vector OA = a, then vector AO = -a.
• Confusing parallel with collinear. Parallel means same direction (or opposite); collinear requires the vectors to also share a common point.
• Arithmetic errors in components. Be careful with negative signs.
• Not simplifying fully. Factor out common scalars to show parallel vectors clearly.

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Edexcel Exam Tips