1.10a, dPureFoundation

Vectors in Two Dimensions

A speed of 30 km/h tells you how fast; a velocity of 30 km/h north tells you how fast $\textit{and which way}$ . That second quantity is a $\textbf{vector}$ , and the whole of this chapter follows from taking the direction seriously — including the fact that you cannot simply add two vectors by adding their sizes.

30 min Vectors

The big picture

Vectors are the language for anything with a direction, which is why this chapter sits at the join between Pure and Mechanics. Every force, velocity and acceleration in the Mechanics strand is a vector, and the rules you meet here are the ones that let you resolve a force into components or add two velocities. Beyond A-Level they are the foundation of linear algebra, computer graphics and physics: the moment a problem has a direction in it, someone reaches for a vector. Getting the notation and the arithmetic automatic now means the Mechanics questions later are about the physics rather than the bookkeeping.

What you'll be able to do

Distinguish vector from scalar quantities and use the standard notations
Add, subtract and scalar-multiply vectors in component and column form
Interpret vector addition geometrically using the triangle and parallelogram rules
Recognise when two vectors are parallel

Vectors, scalars, and how to write them

A $\textbf{scalar}$ has size only — mass, time, distance, speed. A $\textbf{vector}$ has size $\textit{and}$ direction — displacement, velocity, force, acceleration. The pairing matters: distance and displacement are different quantities, and so are speed and velocity.

Three notations appear interchangeably at A-Level. In print a vector is bold, as $\mathbf{a}$ ; handwritten it is underlined, $\underline{a}$ ; and a vector from point $A$ to point $B$ is written $\overrightarrow{AB}$ . In components, $\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}$ and the column form $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ say exactly the same thing, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors along the $x$ - and $y$ -axes.

Two vectors are $\textbf{equal}$ when they have the same magnitude and the same direction. Where they sit on the page is irrelevant — a vector is not tied to a starting point.

\mathbf{a} = a_{1}\mathbf{i} + a_{2}\mathbf{j} = \begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix}

Component form and column form are the same object.

Why this works

Because position is not part of a vector’s identity, two arrows drawn in different places on a diagram can be the same vector. That is what makes translating an arrow around a diagram a legitimate move when adding vectors head-to-tail.

Adding and subtracting: components, then geometry

In components, addition and subtraction work term by term, exactly as you would hope. The geometric picture is what gives the rule meaning: to add $\mathbf{a} + \mathbf{b}$ , draw $\mathbf{b}$ starting where $\mathbf{a}$ finished — the sum is the single arrow from the start of $\mathbf{a}$ to the end of $\mathbf{b}$ . That is the $\textbf{triangle rule}$ .

Subtraction is addition of the reverse. $-\mathbf{b}$ is $\mathbf{b}$ with the same length pointing the opposite way, so $\mathbf{a} - \mathbf{b}$ means $\mathbf{a} + \left(-\mathbf{b}\right)$ .

\begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix} + \begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix} = \begin{pmatrix} a_{1}+b_{1} \\ a_{2}+b_{2} \end{pmatrix}

Add components independently.

\mathbf{a} - \mathbf{b} = \mathbf{a} + \left(-\mathbf{b}\right)

Subtracting reverses the second vector.

Example 1Given

\mathbf{a} = 3\mathbf{i} - \mathbf{j}

and

\mathbf{b} = -\mathbf{i} + 4\mathbf{j}

, find

\mathbf{a} + \mathbf{b}

and

\mathbf{a} - \mathbf{b}

1Sum:

\mathbf{i}

parts

3 + \left(-1\right) = 2

;

\mathbf{j}

parts

-1 + 4 = 3

2Difference:

\mathbf{i}

parts

3 - \left(-1\right) = 4

;

\mathbf{j}

parts

-1 - 4 = -5

Answer

\mathbf{a}+\mathbf{b} = 2\mathbf{i} + 3\mathbf{j}

;

\;\mathbf{a}-\mathbf{b} = 4\mathbf{i} - 5\mathbf{j}

Tip — The magnitudes do not add. $\mathbf{a}$ of length 3 plus $\mathbf{b}$ of length 4 gives a resultant of length 7 only if they point the same way, and length 1 if they oppose.

Scalar multiplication and parallel vectors

Multiplying a vector by a scalar $k$ stretches it by a factor $|k|$ , reversing it if $k$ is negative. In components, every component is multiplied by $k$ .

This gives the test for $\textbf{parallel}$ vectors: $\mathbf{a}$ and $\mathbf{b}$ are parallel exactly when one is a scalar multiple of the other. In practice, check whether the ratio of the $\mathbf{i}$ components equals the ratio of the $\mathbf{j}$ components.

k\begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix} = \begin{pmatrix} ka_{1} \\ ka_{2} \end{pmatrix}

Scale every component.

\mathbf{b} = k\mathbf{a} \iff \mathbf{a} \parallel \mathbf{b}

The parallel test.

Example 1Are

\mathbf{p} = 2\mathbf{i} - 3\mathbf{j}

and

\mathbf{q} = -6\mathbf{i} + 9\mathbf{j}

parallel?

1Compare

\mathbf{i}

components:

-6 \div 2 = -3

2Compare

\mathbf{j}

components:

9 \div \left(-3\right) = -3

3The same scalar works for both, so

\mathbf{q} = -3\mathbf{p}

AnswerYes — parallel, pointing in opposite directions.

Example 2Find

\lambda

such that

\lambda\mathbf{i} + 6\mathbf{j}

is parallel to

2\mathbf{i} + 3\mathbf{j}

1The

\mathbf{j}

components give the scalar:

6 \div 3 = 2

2So the whole vector must be twice the second:

\lambda = 2 \times 2

Answer

\lambda = 4

Why this works

A negative scalar still counts as parallel. Two vectors pointing in exactly opposite directions lie along the same line, which is what "parallel" means for vectors — the sign records the direction along it.

Resultants and the parallelogram rule

The single vector equivalent to several applied in succession is called the $\textbf{resultant}$ . Adding head-to-tail around a chain gives it directly, and the order does not matter — vector addition is commutative, which is exactly why the triangle and parallelogram rules agree.

The parallelogram rule is the same statement drawn differently: place $\mathbf{a}$ and $\mathbf{b}$ tail-to-tail, complete the parallelogram, and the diagonal from the common start is $\mathbf{a}+\mathbf{b}$ . The other diagonal, run from the tip of $\mathbf{b}$ to the tip of $\mathbf{a}$ , is $\mathbf{a}-\mathbf{b}$ .

Example 1A boat is rowed with velocity

4\mathbf{i}

km/h through water flowing at

3\mathbf{j}

km/h. Find the resultant velocity in component form.

1The two velocities add as vectors:

4\mathbf{i} + 3\mathbf{j}

2No component cancels, since one is purely

\mathbf{i}

and the other purely

\mathbf{j}

Answer

4\mathbf{i} + 3\mathbf{j}

km/h

Tip — If a chain of vectors returns to where it began, the resultant is the $\textbf{zero vector}$ $\mathbf{0}$ — not the number 0. Writing $\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$ instead of $\mathbf{0}$ is a small notation slip examiners do notice.

Think like an examiner

Keep the notation consistent: bold or underlined for vectors, and never mix $\mathbf{i}$ / $\mathbf{j}$ form with column form inside a single line of working.
Give answers in the form the question uses. If it sets up in column vectors, answer in column vectors.
When asked to show two vectors are parallel, state the scalar explicitly — " $\mathbf{q} = -3\mathbf{p}$ , so they are parallel" is what scores.
The zero vector is $\mathbf{0}$ , written bold. It is a vector, not the scalar zero.

Common misconceptions

Adding magnitudes:

|\mathbf{a}| = 3

|\mathbf{b}| = 4

, so

|\mathbf{a}+\mathbf{b}| = 7

Only true if they point the same way. Add the components first, then find the magnitude of the result.

Treating

\mathbf{a} - \mathbf{b}

and

\mathbf{b} - \mathbf{a}

as the same.

They have equal magnitude but opposite direction:

\mathbf{b}-\mathbf{a} = -\left(\mathbf{a}-\mathbf{b}\right)

Concluding two vectors are not parallel because the scalar is negative.

A negative scalar still means parallel — the vectors lie on the same line, pointing opposite ways.

Writing a vector answer without bold or underline.

Notation is part of the answer.

3\mathbf{i}+4\mathbf{j}

is a vector;

3i+4j

read as scalars is not.

Vector basics

\mathbf{a} = a_{1}\mathbf{i}+a_{2}\mathbf{j}

Component form.

\mathbf{a}+\mathbf{b} = \left(a_{1}+b_{1}\right)\mathbf{i}+\left(a_{2}+b_{2}\right)\mathbf{j}

Add componentwise.

k\mathbf{a} = ka_{1}\mathbf{i}+ka_{2}\mathbf{j}

Scalar multiplication.

\mathbf{b}=k\mathbf{a} \iff \text{parallel}

The parallel test.

Stretch yourself

The vectors $\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}$ and $\mathbf{b} = 5\mathbf{i} - \mathbf{j}$ are given. Find scalars $s$ and $t$ such that $s\mathbf{a} + t\mathbf{b} = 16\mathbf{i} + 7\mathbf{j}$ .

Hint — Components are independent — compare the $\mathbf{i}$ parts and the $\mathbf{j}$ parts separately to get two equations.

Questions students ask

What is the difference between distance and displacement?

Distance is a scalar — total path length travelled. Displacement is a vector — the straight-line change in position from start to finish. Walk a full lap of a track and the distance is 400 m while the displacement is

\mathbf{0}

Does it matter whether I use column vectors or

\mathbf{i}

\mathbf{j}

No, they are equivalent and both score. Follow the form the question uses, and stay in one form within a piece of working.

Can a vector have a negative magnitude?

No. Magnitude is a length, so it is always zero or positive. A negative scalar multiple reverses direction; it does not create a negative length.

Are

\overrightarrow{AB}

and

\overrightarrow{BA}

the same vector?

No — they are negatives of each other. Same length, opposite direction, so

\overrightarrow{BA} = -\overrightarrow{AB}

Key takeaways

A vector has magnitude and direction; a scalar has magnitude only.
Component form $a_{1}\mathbf{i}+a_{2}\mathbf{j}$ and column form are the same object.
Add and subtract componentwise; geometrically this is the triangle rule.
Scalar multiplication scales length and may reverse direction.
Two vectors are parallel exactly when one is a scalar multiple of the other, negative scalars included.

How this fits the course

Magnitude and Direction

Leads to

Magnitude and Direction

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 5 lessons

Vectors in Two Dimensions

A speed of 30 km/h tells you how fast; a velocity of 30 km/h north tells you how fast

\textit{and which way}

. That second quantity is a

\textbf{vector}

, and the whole of this chapter follows from taking the direction seriously — including the fact that you cannot simply add two vectors by adding their sizes.

30 min Vectors