Vectors

Q: What is the difference between a vector and a scalar?

A scalar has only magnitude (e.g. length, mass) while a vector has both magnitude and direction (e.g. displacement, force).

Q: Do I get the vector magnitude formula in the exam?

No, you must recall that the magnitude of a column vector \text{\textbackslash}end{pmatrix} uses Pythagoras' theorem: .

Edexcel International GCSE Mathematics A· Section 5.1· 25 min read

1. Vectors: Definition and Standard Notation★★☆☆☆⏱ 5 min

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📘 Definition

Vector

A quantity that has both magnitude (size) and direction, unlike a scalar which only has magnitude.

Example:

Displacement (distance plus direction) is a vector; distance alone is a scalar.

Three standard notations are used for vectors in your exam: bold lowercase letters ( $a$ , $b$ ), arrow notation for position vectors ( $\textbackslash$ vec{OA} $f or t h e v ec t or f r o m or i g in O t o p o in t A), an d * * co l u mn v ec t or s * * w r i tt e na s$ \begin{pmatrix}x \ y $\ e n d p ma t r i x w h er e$ x $i s t h e h or i z o n t a l co m p o n e n t an d$ y$ the vertical component.

📐 Worked Example

Write the vector representing a displacement of 4 units right and 2 units down in column vector form, and name two other valid notations for this vector.

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1. Horizontal component: right = positive $x$ , so $x = 4$ .
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1. Vertical component: down = negative $y$ , so $y = - 2$ .
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$\begin{pmatrix}4 \\ -2\end{pmatrix}$
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1. Other valid notations: bold $v$ or $\textbackslash$ vec{OP} $w h er e$ P $i s t h e p o in t (4, - 2) r e l a t i v e t oor i g in$ O$.

2. Vector Addition, Subtraction and Scalar Multiplication★★★☆☆⏱ 7 min

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📘 Definition

Scalar Multiple

$k a$

The vector formed by multiplying vector $a$ by scalar $k$ . If $k < 0$ , the direction of the vector is reversed, and magnitude scales by $∣ k ∣$ .

Example:

If $\textbf{a} = \begin{pmatrix}2 \ 3$ \text{\textbackslash}end{pmatrix}, then $3\textbf{a} = \begin{pmatrix}6 \ 9$ \text{\textbackslash}end{pmatrix} and $-2\textbf{a} = \begin{pmatrix}-4 \ -6$ \text{\textbackslash}end{pmatrix}.

To add or subtract column vectors, add or subtract corresponding components. For geometric vectors, use the triangle law: place the tail of the second vector at the head of the first, then the resultant connects the tail of the first to the head of the second.

📐 Worked Example

Given $a = (3 - 1)$ and $b = (- 2 5)$ , calculate $2 a - 3 b$ .

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1. Calculate $2 a$ by multiplying each component of $a$ by 2:
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$2\mathbf{a} = \begin{pmatrix}2 \times 3 \\ 2 \times (-1)\end{pmatrix} = \begin{pmatrix}6 \\ -2\end{pmatrix}$
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1. Calculate $3 b$ by multiplying each component of $b$ by 3:
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$3\mathbf{b} = \begin{pmatrix}3 \times (-2) \\ 3 \times 5\end{pmatrix} = \begin{pmatrix}-6 \\ 15\end{pmatrix}$
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1. Subtract $3 b$ from $2 a$ by subtracting corresponding components:
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$2\mathbf{a} - 3\mathbf{b} = \begin{pmatrix}6 - (-6) \\ -2 - 15\end{pmatrix} = \begin{pmatrix}12 \\ -17\end{pmatrix}$

3. Calculating the Modulus (Magnitude) of a Vector★★☆☆☆⏱ 4 min

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The magnitude (or modulus) of a vector is its length, written $∣ a ∣$ for vector $a$ . For a column vector $\begin{pmatrix}x \ y$ \text{\textbackslash}end{pmatrix}, use Pythagoras' theorem to calculate magnitude, as the components form the two shorter sides of a right-angled triangle.

|\mathbf{v}| = \sqrt{x^2 + y^2}

📐 Worked Example

Calculate the magnitude of the vector $v = (5 - 3)$ , giving your answer to 2 decimal places.

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1. Identify $x = 5$ , $y = - 3$ .
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1. Square each component: $x^{2} = 5^{2} = 25$ , $y^{2} = (- 3)^{2} = 9$ .
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1. Sum the squares: $25 + 9 = 34$ .
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$|\mathbf{v}| = \sqrt{34} \approx 5.83$

4. Resultant Vectors and Simple Geometric Proofs★★★★☆⏱ 9 min

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A resultant vector is the sum of two or more vectors, representing a single equivalent journey. For geometric problems, follow the path of given vectors to write expressions for unknown vectors: if you travel opposite to the direction of a given vector, multiply it by $- 1$ . Parallel vectors are scalar multiples of each other; collinear points lie on the same straight line, so the vector between them is a scalar multiple of another vector along the line.

📐 Worked Example

In shape OABC, $\textbackslash$ vec{OA} = 3 $a$ , $\textbackslash$ vec{AB} = 2 $b$ , $\textbackslash$ vec{BC} = $c$ . Find (a) $\textbackslash$ vec{OC} $, (b)$ \textbackslash $v ec C A$ . Prove that AC is parallel to $d = c + 2 b$ .

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1a. Find $\textbackslash$ vec{OC} $b y f o l l o w in g t h e p a t h$ O \rightarrow A \rightarrow B \rightarrow C$:
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$\vec{OC} = \vec{OA} + \vec{AB} + \vec{BC} = 3\mathbf{a} + 2\mathbf{b} + \mathbf{c}$
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1b. Find $\textbackslash$ vec{CA} $b y f o l l o w in g t h e p a t h$ C \rightarrow B \rightarrow A$, reversing vector signs for travel against their defined direction:
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$\vec{CA} = \vec{CB} + \vec{BA} = -\vec{BC} - \vec{AB} = -\mathbf{c} - 2\mathbf{b}$
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1. Prove parallelism: rewrite $\textbackslash$ vec{CA} $a s a sc a l a r m u l t i pl eo f$ \textbf{d}$:
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$\vec{CA} = -1(\mathbf{c} + 2\mathbf{b}) = -1\mathbf{d}$
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1. Conclusion: Since $\textbackslash$ vec{CA} $i s a sc a l a r m u l t i pl eo f$ \textbf{d}$, the two vectors are parallel.

5. Common Pitfalls

Wrong move:

Writing vectors without bold or arrow notation, e.g. writing $a$ instead of $a$ or $\textbackslash$ vec{a}$.

Why:

Examiners mark unmarked letters as scalars, so you will lose marks for incorrect notation.

Correct move:

Always use bold lower case for named vectors, or arrow notation for position vectors between two points.

Wrong move:

Making the vertical component of a column vector positive when displacement is down, or negative when up.

Why:

Standard convention uses positive $y$ for upwards direction, so incorrect signs give the wrong vector direction.

Correct move:

Follow coordinate rules: right = + $x$ , left = - $x$ , up = + $y$ , down = - $y$ .

Wrong move:

Calculating magnitude as $\textbackslash$ sqrt{x + y} $in s t e a d o f$ \textbackslash $s q r t x^{2} + y^{2}$ , or incorrectly squaring negative components.

Why:

Magnitude uses Pythagoras' theorem, so incorrect squaring or addition gives a wrong length value.

Correct move:

Always square both components first, sum them, then take the positive square root.

Wrong move:

Adding vectors by adding their magnitudes directly, ignoring direction.

Why:

Vectors have direction, so magnitudes only add if vectors point in exactly the same direction.

Correct move:

Add vectors component-wise for column vectors, or use the triangle law for geometric vectors.

Wrong move:

Forgetting to reverse the sign of a vector when travelling against its defined direction in geometric problems.

Why:

Travelling from $B$ to $A$ is the opposite of travelling from $A$ to $B$ , so the vector sign must be flipped.

Correct move:

If you travel opposite to the arrow of a given vector, multiply it by $- 1$ before adding to your resultant.

6. Quick Reference Cheatsheet

Concept	Formula/Rule	Example
Vector Notation	Bold $a$ / $\textbackslash$ vec{OA} $/$ \begin{pmatrix}x \ y$\text{\textbackslash}end{pmatrix}	2 right, 3 up = $\begin{pmatrix}2 \ 3$ \text{\textbackslash}end{pmatrix}
Scalar Multiplication	$k\begin{pmatrix}x \ y$ \text{\textbackslash}end{pmatrix} = $\begin{pmatrix}kx \ ky$ \text{\textbackslash}end{pmatrix}	$2\begin{pmatrix}1 \ 4$ \text{\textbackslash}end{pmatrix} = $\begin{pmatrix}2 \ 8$ \text{\textbackslash}end{pmatrix}
Vector Addition	$\begin{pmatrix}x_1 \ y_1$ \text{\textbackslash}end{pmatrix} + $\begin{pmatrix}x_2 \ y_2$ \text{\textbackslash}end{pmatrix} = $\begin{pmatrix}x_1+x_2 \ y_1+y_2$ \text{\textbackslash}end{pmatrix}	$\begin{pmatrix}3 \ 1$ \text{\textbackslash}end{pmatrix} + $\begin{pmatrix}-2 \ 4$ \text{\textbackslash}end{pmatrix} = $\begin{pmatrix}1 \ 5$ \text{\textbackslash}end{pmatrix}
Magnitude (Modulus)	$\|\begin{pmatrix}x \ y$ \text{\textbackslash}end{pmatrix}\| = $\textbackslash$ sqrt{x^2 + y^2}$	$\|\begin{pmatrix}3 \ -4$ \text{\textbackslash}end{pmatrix}\| = 5
Resultant Vectors	Sum of vectors along a path, flip sign if travelling opposite direction	$\textbackslash$ vec{AC} = $\textbackslash$ vec{AB} + $\textbackslash$ vec{BC}$
Parallel Vectors	Vectors are scalar multiples of each other	If $a = 2 b$ , $a ∣∣ b$

7. Frequently Asked

What is the difference between a vector and a scalar?

A scalar has only magnitude (e.g. length, mass) while a vector has both magnitude and direction (e.g. displacement, force).

Do I get the vector magnitude formula in the exam?

No, you must recall that the magnitude of a column vector $\begin{pmatrix}x \ y$ \text{\textbackslash}end{pmatrix} uses Pythagoras' theorem: $\textbackslash s q r t x^{2} + y^{2}$ .

What's Next

Now you have mastered core vector concepts for Edexcel IGCSE Maths A Higher Tier, you can apply this knowledge to transformation geometry, where column vectors are used to describe translations of shapes on coordinate grids. You will also encounter vectors in high-mark problem-solving questions combining geometry and algebra, which often appear in Paper 2H and 4H. Make sure you practise past paper vector proof questions to get comfortable with justifying conclusions using vector rules, as many students lose marks here due to incomplete justification. Remember this topic is Higher Tier only, so Foundation students do not need to revise it.