Pure 1 — Vectors | Full Lesson

Pure Mathematics 1

Vectors Lesson

Hello. I am going to teach you everything you need to know about vectors. I will assume you know nothing — and by the end, you will be solving vector geometry problems with confidence. Let us begin.

What is a Vector?

A vector is a quantity that has both magnitude (size) and direction. This is different from a scalar, which only has magnitude.

Scalar vs Vector Scalar: speed, mass, temperature | Vector: velocity, force, displacement

Think of it this way: "5 km" is a scalar (just a distance). But "5 km north" is a vector — it tells you how far and in which direction.

Vector Notation

There are several ways to write a vector:

Bold: v | Underline: v | Arrow: AB with arrow | Column: (3 over 4)

In A-level maths, you will most often see bold or the column vector form. The vector from point A to point B is written as AB with an arrow above.

Example 1: Identifying Scalars and Vectors

Which of these are scalars and which are vectors?

a) 30 km/h b) 30 km/h due east c) 10 Newtons downward d) 5 kg

a) Scalar — speed has no direction

b) Vector — velocity has direction (due east)

c) Vector — force has direction (downward)

d) Scalar — mass has no direction

Scalars: a, d | Vectors: b, c

A vector can be represented as a directed line segment — an arrow drawn from one point to another. The length of the arrow shows the magnitude, and the arrowhead shows the direction.

Teacher's Tip: Every time you see a quantity, ask yourself: "Does it have a direction?" If the answer is yes, it is a vector. If no, it is a scalar. This is the simplest test you will ever learn.

Position Vectors

A position vector tells you the location of a point relative to the origin O. If a point A has coordinates (x, y), then its position vector is written as a or OA.

Position Vector OA = (x, y) = xi + yj

The vector from the origin to point A (x, y) is simply the column vector with those coordinates. We can also write it using i and j notation (which we will cover properly in section 6).

If we have two points A and B, the vector from A to B is:

Vector Between Two Points AB = b − a

This makes sense: to go from A to B, you go from A back to the origin (−a), then from the origin to B (+b). So AB = b − a.

Example 2: Position Vectors

Point A is at (3, 7) and point B is at (5, 2). Find a, b, and AB.

a = (3, 7) → a = 3i + 7j

b = (5, 2) → b = 5i + 2j

AB = b - a = (5-3)i + (2-7)j = 2i - 5j

Answer: AB = (2, −5) or 2i − 5j

Teacher's Tip: "Position vector" is just a fancy name for "the coordinates written as a vector." Do not overthink it. The key formula is AB = b − a. Memorise this.

Vector Addition and Subtraction

Adding and subtracting vectors is simple: you just add or subtract the corresponding components.

Vector Addition (a₁, a₂) + (b₁, b₂) = (a₁ + b₁, a₂ + b₂)

For subtraction, you do the same but subtract instead of adding.

Triangle Law (Head-to-Tail)

To add vectors geometrically: draw the first vector, then draw the second vector starting from the tip of the first. The resultant vector goes from the tail of the first to the tip of the second.

Example 3: Vector Addition

Given a = 3i + 2j and b = i − 4j, find a + b and a − b.

a + b = (3+1)i + (2-4)j = 4i - 2j

a - b = (3-1)i + (2-(-4))j = 2i + 6j

a + b = 4i − 2j | a − b = 2i + 6j

Parallelogram Law

An alternative way: draw both vectors from the same starting point, then complete the parallelogram. The diagonal from the common start to the opposite corner is the resultant vector.

Vector addition is commutative: a + b = b + a. The order does not matter.

Teacher's Tip: Head-to-tail is the most intuitive method. Just remember: the resultant always goes from the starting tail to the final tip. Try the interactive visualiser below to see this in action.

Vector Addition Visualiser

a b a + b

a: x 3 y 2 b: x 1 y -4

a = 3i + 2j b = 1i − 4j a + b = 4i − 2j

Scalar Multiplication

Scalar multiplication means multiplying a vector by a number (a scalar). Every component of the vector is multiplied by that number.

Scalar Multiplication k(x, y) = (kx, ky)

If k > 0, the vector keeps its direction but its magnitude is scaled by k. If k < 0, the vector reverses direction.

Example 4: Scalar Multiplication

Given v = 3i + 2j, find 2v and −3v.

2v = 2(3i + 2j) = 6i + 4j

-3v = -3(3i + 2j) = -9i - 6j

2v = 6i + 4j | −3v = −9i − 6j

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other. In other words, a is parallel to b if there exists a scalar k such that a = kb.

If k > 0, the vectors point in the same direction. If k < 0, they point in opposite directions (but are still parallel).

Teacher's Tip: Scalar multiplication is the simplest operation — you just multiply every component. The real power is that it lets you recognise parallel vectors instantly.

Magnitude of a Vector

The magnitude (or length) of a vector tells you how big it is. For a vector v = xi + yj, the magnitude is written |v| and is found using Pythagoras' theorem.

Magnitude of a Vector |v| = √(x² + y²)

This is exactly the same as the distance formula. If v = AB where A = (x₁, y₁) and B = (x₂, y₂), then:

Distance Formula (same thing) |AB| = √((x₂ − x₁)² + (y₂ − y₁)²)

The magnitude is always a positive number (or zero for the zero vector).

Example 5: Finding Magnitude

Find |v| for v = 6i + 8j.

|v| = √(6² + 8²) = √(36 + 64) = √100 = 10

|v| = 10

Example 6: Distance Between Two Points

Find the distance between A(1, 3) and B(5, 6).

AB = (5-1)i + (6-3)j = 4i + 3j

|AB| = √(4² + 3²) = √(16 + 9) = √25 = 5

Distance = 5 units

Teacher's Tip: Magnitude is just Pythagoras. If you can find the hypotenuse of a right-angled triangle, you can find the magnitude of a vector. The two ideas are identical.

Unit Vectors and i, j Notation

A unit vector is a vector with magnitude exactly 1. It simply shows direction without any size.

In 2D, the two most important unit vectors are:

Standard Unit Vectors i = (1, 0) j = (0, 1)

i points along the x-axis (to the right) and j points along the y-axis (upwards). Any 2D vector can be written as a combination of these two:

Any Vector in i, j Form (x, y) = xi + yj

Finding a Unit Vector in Any Direction

To find a unit vector in the same direction as a given vector v, divide v by its magnitude:

Unit Vector Formula û = v / |v| = (1/|v|) × (x, y)

The hat notation û means "unit vector in the direction of v."

Example 7: Finding a Unit Vector

Find the unit vector in the direction of v = 3i + 4j.

Step 1: Find magnitude: |v| = √(3² + 4²) = √25 = 5

Step 2: Divide by magnitude:

û = (3/5)i + (4/5)j

Unit vector = (3/5)i + (4/5)j = 0.6i + 0.8j

Teacher's Tip: The unit vector is just the original vector "shrunk" to length 1 while keeping the same direction. Divide each component by the magnitude — that is all there is to it.

Parallel Vectors and Collinearity

Parallel Vectors

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

a = (2, 4) and b = (6, 12) are parallel because b = 3a.

a = (3, 6) and b = (−1, −2) are parallel because a = −3b.

Parallel Condition a ∥ b ⇔ a = kb for some scalar k

To check if two vectors are parallel, compare the ratios of their components. If x₁/x₂ = y₁/y₂, they are parallel.

Collinearity

Three points A, B, and C are collinear (lie on the same straight line) if the vectors AB and AC are parallel. In other words, one is a scalar multiple of the other.

Example 8: Proving Collinearity

Show that points A(1, 2), B(3, 5), and C(7, 11) are collinear.

Step 1: Find AB:

AB = (3-1)i + (5-2)j = 2i + 3j

Step 2: Find AC:

AC = (7-1)i + (11-2)j = 6i + 9j

Step 3: Check: AC = 3 × (2i + 3j) = 3 AB

Since AC = 3 × AB, the vectors are parallel. A, B, and C share point A, so they lie on the same line.

A, B, C are collinear ✓

Teacher's Tip: Collinearity is just parallel vectors with a shared point. If two vectors from the same point are parallel, all three points lie on a straight line. Always show the scalar relationship clearly.

Solving Vector Geometry Problems

Vectors are incredibly powerful for solving geometry problems. You can find midpoints, divide lines in ratios, and prove geometric facts without drawing a diagram.

Midpoint

The midpoint M of points A and B has position vector:

Midpoint Formula m = (a + b) / 2

Example 9: Finding a Midpoint

Find the midpoint of A(2, 5) and B(6, 9).

a = 2i + 5j, b = 6i + 9j

m = (a + b)/2 = ((2+6)i + (5+9)j)/2 = (8i + 14j)/2

m = 4i + 7j

M(4, 7)

Dividing a Line in a Given Ratio

If point P divides AB in the ratio m : n, then:

Section Formula p = (na + mb) / (m + n)

Think of it as a weighted average — the weight is the opposite part of the ratio.

Example 10: Dividing in a Ratio

Point P divides AB in the ratio 2 : 3 where A(1, 2) and B(6, 7). Find P.

m = 2, n = 3 (P is 2 parts from A, 3 parts from B)

p = (3a + 2b) / (2+3)

p = (3(1,2) + 2(6,7)) / 5

p = ((3+12), (6+14)) / 5 = (15, 20) / 5

p = (3, 4)

P(3, 4)

Proving Geometric Properties

You can use vectors to prove things like "the line joining midpoints of two sides of a triangle is parallel to the third side and half its length."

Example 11: Midpoint Theorem

In triangle ABC, M is the midpoint of AB and N is the midpoint of AC. Prove that MN = (1/2) × BC.

Let the position vectors be a, b, c.

M = (a + b)/2 (midpoint of AB)

N = (a + c)/2 (midpoint of AC)

MN = N - M = (a + c)/2 - (a + b)/2

= (a + c - a - b)/2 = (c - b)/2

= \frac{1}{2}(c - b) = \frac{1}{2}BC

Therefore MN = (1/2) × BC ✓
(parallel and half the length)

Teacher's Tip: Vector geometry proofs follow a pattern: assign position vectors, express everything in terms of them, then simplify. The algebra does the work for you — no diagrams needed.

Practice — Step by Step

Work through each vector problem one step at a time. Type your answer for each step and click Check Step. You must get each step correct before moving to the next.

Mastery Kit

Vectors are not just arrows; they are the language in which geometry speaks to algebra.

— Adapted from a mathematical aphorism

⚡ Quick Hacks

Vector addition shortcut: Just add corresponding components. x to x, y to y. No cross-terms. It is that simple.
Magnitude trick: |v| = √(x² + y²). If you recognise a Pythagorean triple (3-4-5, 5-12-13, etc.), you can find the magnitude instantly without a calculator.
Collinearity check: Show AB = k × AC for some scalar k. If the ratio works, they are collinear. No diagram needed.
Unit vector memory: "Divide and conquer" — divide each component by the magnitude. The result always has length 1.
Midpoint = average: The midpoint is just the average of the two position vectors. (a + b)/2.

📌 Key Notes

AB = b − a: This is the most important vector formula you will learn. The vector from A to B is the position vector of B minus the position vector of A.
Scalar multiple = parallel: If a = kb for any k (positive or negative), a and b are parallel. This is the definition.
Zero vector: (0, 0) has magnitude 0 and no direction. Adding it to any vector does nothing.
Negative vector: −v has the same magnitude as v but points in the exact opposite direction.
Triangle law: a + b = c means the three vectors form a triangle when placed head-to-tail.

⚠️ Common Mistakes

Adding wrong components: Always add x to x and y to y. Never mix them. x + y makes no sense.
Forgetting to subtract for AB: AB = b − a, not a − b. The order matters: "from A to B" means B minus A.
Magnitude sign errors: x² + y² is always positive even if x or y is negative. The magnitude is always non-negative.
Calling i and j "unit vectors" wrong: i = (1, 0) and j = (0, 1). They have magnitude 1. Any other vector divided by its magnitude is also a unit vector.
Ratio confusion: For AB divided in ratio m:n, the weight for a is n (the opposite part). Many students get this backwards.

🧠 Memory Aid — Vector Cheat Sheet

Vector: v = xi + yj or (x, y)

Magnitude: |v| = √(x² + y²)

Unit vector: û = v / |v|

AB = b − a

Parallel: a = kb for some scalar k

Midpoint: m = (a + b)/2

Section formula: p = (na + mb)/(m+n)

💡

The Golden Rule of Vectors

A vector is just a journey. The components tell you how far to travel in each direction. Add components, multiply components, find the hypotenuse — that is all vectors ever ask you to do. Master these and you master vectors.

You Have Completed the Vectors Lesson

Here is what you have learned:

✅ What a vector is (magnitude + direction) vs a scalar
✅ Position vectors and AB = b − a
✅ Vector addition and subtraction (head-to-tail, triangle law)
✅ Scalar multiplication: kv = (kx, ky)
✅ Magnitude: |v| = √(x² + y²)
✅ Unit vectors and i, j notation
✅ Parallel vectors and collinearity
✅ Vector geometry: midpoints, ratios, and geometric proofs

Well done. Vectors are a foundation topic — they appear everywhere in mechanics, geometry, and further maths. You now have the toolkit.