Vectors

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Notes

Introduction to Column Vectors

A **column vector** describes a translation: e.g. $\begin{pmatrix}6\\3\end{pmatrix}$ means 6 right, 3 up.
Add/subtract vectors component-wise: top numbers together, bottom numbers together.
Multiply a vector by a **scalar** (a number) by multiplying each component.
Follow order of operations when combining scalar multiplication and addition.
Example: $2\begin{pmatrix}5\\2\end{pmatrix}+5\begin{pmatrix}3\\-1\end{pmatrix}=\begin{pmatrix}10\\4\end{pmatrix}+\begin{pmatrix}15\\-5\end{pmatrix}=\begin{pmatrix}25\\-1\end{pmatrix}$ .

Representing Vectors as Diagrams

A vector has **magnitude** (size) and **direction**; shown by an arrow.
Vectors are written in bold (or underlined by hand), e.g. $\mathbf{a}$ or $\underline{a}$ .
$\overrightarrow{AB}$ goes from A to B; $\overrightarrow{BA}$ is opposite direction.
To draw $\begin{pmatrix}3\\4\end{pmatrix}$ , start at a point, move 3 right, 4 up, draw arrow.
Multiplying by a positive scalar changes length but not direction; negative scalar reverses direction.
To add vectors diagrammatically, place them tip-to-tail; the resultant goes from start to end.

Magnitude of a Vector

The **magnitude** (modulus) of a vector is its length, always positive.
For $\mathbf{a}=\begin{pmatrix}x\\y\end{pmatrix}$ , $|\mathbf{a}|=\sqrt{x^2+y^2}$ (Pythagoras).
Magnitude is independent of direction: $|\overrightarrow{AB}|=|\overrightarrow{BA}|$ .
If a vector is multiplied by scalar $k$ , its magnitude is multiplied by $|k|$ .

Position & Displacement Vectors

A **position vector** locates a point relative to the origin O: $\overrightarrow{OA}=\mathbf{a}$ .
Coordinates equal components: point (3,-2) has position vector $\begin{pmatrix}3\\-2\end{pmatrix}$ .
A **displacement vector** goes from one point to another: $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$ .
Use $\overrightarrow{AB}=-\mathbf{a}+\mathbf{b}$ to find displacement from position vectors.

Finding Vector Paths

A **vector path** is a sequence of vectors from start to end.
In a grid of parallelograms, horizontal moves are multiples of one vector, diagonal moves of another.
Count steps in each direction to express a vector in terms of given vectors.
Negative signs indicate opposite direction.

Problem Solving with Vectors

Two vectors are **parallel** if one is a scalar multiple of the other.
To prove three points are **collinear**, show two vectors between them are parallel and share a common point.
Ratios along a line: if $AX:XB=3:5$ , then $\overrightarrow{AX}=\frac{3}{8}\overrightarrow{AB}$ .
Use vector algebra to find unknown points or prove geometric properties.

Column Vector Example

Vector Addition (Tip-to-Tail)

Magnitude of a Vector

Position and Displacement Vectors

Practice questions

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1.What is the column vector representing a translation of 3 units to the right and 4 units down?
Easy
- A $\begin{pmatrix}3\\-4\end{pmatrix}$
- B $\begin{pmatrix}-3\\4\end{pmatrix}$
- C $\begin{pmatrix}3\\4\end{pmatrix}$
- D $\begin{pmatrix}-3\\-4\end{pmatrix}$
2.Given $\mathbf{a} = \begin{pmatrix}5\\2\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}3\\-1\end{pmatrix}$ , find $\mathbf{a} + \mathbf{b}$ .
Easy
- A $\begin{pmatrix}8\\1\end{pmatrix}$
- B $\begin{pmatrix}8\\3\end{pmatrix}$
- C $\begin{pmatrix}2\\1\end{pmatrix}$
- D $\begin{pmatrix}2\\3\end{pmatrix}$
3.Given $\mathbf{a} = \begin{pmatrix}4\\-2\end{pmatrix}$ , what is $2\mathbf{a}$ ?
Easy
- A $\begin{pmatrix}8\\-4\end{pmatrix}$
- B $\begin{pmatrix}6\\0\end{pmatrix}$
- C $\begin{pmatrix}8\\4\end{pmatrix}$
- D $\begin{pmatrix}2\\-1\end{pmatrix}$
4.Find the magnitude of the vector $\begin{pmatrix}3\\4\end{pmatrix}$ .
Easy
- A5
- B7
- C12
- D25
5.The displacement vector from A to B is $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ . If $\mathbf{a} = \begin{pmatrix}2\\5\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}7\\1\end{pmatrix}$ , what is $\overrightarrow{AB}$ ?
Easy
- A $\begin{pmatrix}5\\-4\end{pmatrix}$
- B $\begin{pmatrix}5\\4\end{pmatrix}$
- C $\begin{pmatrix}-5\\4\end{pmatrix}$
- D $\begin{pmatrix}9\\6\end{pmatrix}$
6.Given $\mathbf{p} = \begin{pmatrix}4\\5\end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix}-2\\7\end{pmatrix}$ , find $2\mathbf{p} + \mathbf{q}$ .
Medium
- A $\begin{pmatrix}6\\17\end{pmatrix}$
- B $\begin{pmatrix}6\\-3\end{pmatrix}$
- C $\begin{pmatrix}2\\12\end{pmatrix}$
- D $\begin{pmatrix}10\\12\end{pmatrix}$
7.Given $\mathbf{a} = \begin{pmatrix}-3\\2\end{pmatrix}$ , find $|\mathbf{a}|$ .
Medium
- A $\sqrt{13}$
- B $\sqrt{5}$
- C13
- D5
8.Point A is (6,4) and point B is (2,7). Write $\overrightarrow{AB}$ as a column vector.
Medium
- A $\begin{pmatrix}-4\\3\end{pmatrix}$
- B $\begin{pmatrix}4\\-3\end{pmatrix}$
- C $\begin{pmatrix}-4\\-3\end{pmatrix}$
- D $\begin{pmatrix}4\\3\end{pmatrix}$

Summary

## Vectors - IGCSE Maths (CIE) Extended ### Column Vectors - A column vector describes a translation: $\begin{pmatrix} x \\ y \end{pmatrix}$ means $x$ units right, $y$ units up. - **Addition**: Add top and bottom numbers separately: $\begin{pmatrix} 5 \\ 2 \end{pmatrix} + \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 8 \\ 1 \end{pmatrix}$. - **Subtraction**: Subtract components: $\begin{pmatrix} 5 \\ 2 \end{pmatrix} - \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$. - **Scalar multiplication**: Multiply each component: $3\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}$. ### Magnitude (Modulus) - Length of vector $\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}$: $|\mathbf{a}| = \sqrt{x^2 + y^2}$. - Always positive; direction irrelevant. ### Position & Displacement Vectors - **Position vector** of point A from origin O: $\overrightarrow{OA} = \mathbf{a}$. - **Displacement vector** from A to B: $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$. ### Vector Paths - Follow paths using given vectors (e.g., $\mathbf{a}$, $\mathbf{b}$). - Opposite direction: $-\mathbf{a}$. - Simplify by combining like terms. ### Problem Solving - **Parallel vectors**: One is scalar multiple of the other: $\mathbf{b} = k\mathbf{a}$. - **Collinearity**: Show two vectors are parallel and share a common point. - **Ratios**: Convert to fractions. E.g., $AP:PC = 3:1$ means $\overrightarrow{AP} = \frac{3}{4}\overrightarrow{AC}$. ### Key Formulas - $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ - $|\mathbf{a}| = \sqrt{x^2 + y^2}$ - Parallel: $\mathbf{b} = k\mathbf{a}$

Key terms

Column Vector: A vector written as $\begin{pmatrix} x \\ y \end{pmatrix}$ representing horizontal and vertical components.
Magnitude: The length of a vector, found using Pythagoras: $\sqrt{x^2 + y^2}$.
Position Vector: The vector from the origin O to a point, e.g., $\overrightarrow{OA} = \mathbf{a}$.
Displacement Vector: The vector from one point to another: $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$.
Scalar: A number that multiplies a vector, changing its magnitude but not direction (unless negative).
Parallel Vectors: Vectors that are scalar multiples of each other: $\mathbf{b} = k\mathbf{a}$.
Collinear: Points that lie on the same straight line; shown by parallel vectors sharing a point.

Slides

Introduction to Column Vectors

A column vector describes a translation from one point to another.
For example, ⎛6⎞ means 6 units right and 3 units up.
Vectors are written in bold or with an underline, e.g., a or a.

What does the column vector ⎛4⎞ represent?

4 units right, 2 units down
4 units right, 2 units up
4 units left, 2 units up
4 units left, 2 units down

Adding and Subtracting Column Vectors

Add vectors by adding corresponding components: ⎛5⎞ + ⎛3⎞ = ⎛8⎞.
Subtract vectors by subtracting components: ⎛5⎞ - ⎛3⎞ = ⎛2⎞.
Work with top and bottom numbers separately.

If $a =$ ⎛2⎞ and $b =$ ⎛1⎞, then $a + b =$ ⎛____⎞.

Multiplying a Vector by a Scalar

A scalar is a number (not a vector).
Multiply each component by the scalar: 3⎛2⎞ = ⎛6⎞.
If the scalar is negative, the direction reverses.

What is 2⎛3⎞?

⎛6⎞
⎛5⎞
⎛6⎞
⎛3⎞

Magnitude of a Vector

The magnitude (or modulus) is the length of the vector.
For $a =$ ⎛x⎞, $|a| = \sqrt{x^{2} + y^{2}}$ .
It is always positive and ignores direction.

The magnitude of vector ⎛3⎞ $is \sqrt{3^{2} + ____^{2}} = √____$ .

Position and Displacement Vectors

A position vector describes a point relative to the origin $O: a = OA$ .
A displacement vector from A to B is $AB = b - a$ .
$AB = -a + b = b - a$ .

Which of these are true?

$AB = b - a$
$AB = a - b$
The position vector of (3, -2) is ⎛3⎞

Finding Vector Paths

A vector path is a route from start to end using given vectors.
Count how many times each vector is used in the correct direction.
If a vector goes opposite direction, use a negative sign.

In a grid of parallelograms with $a = AB$ and $b = AF$ , what is AE if it is 4 steps right?

4a
4b
$a + 4b$
$4a + b$

Parallel Vectors and Collinearity

Two vectors are parallel if one is a scalar multiple of the other: $b = ka$ .
Three points A, B, C are collinear if AB and BC are parallel and share point B.
If $AB = k BC$ , then A, B, C lie on a straight line.

Which of these are true?

Vectors ⎛2⎞ and ⎛6⎞ are parallel.
If $AB = 2 BC$ , then A, B, C are collinear.
A negative scalar multiple reverses direction.

Using Ratios in Vector Problems

Convert ratios to fractions: if $AP:PB = 3:5$ , then $AP = \frac{3}{8} AB$ .
Always check which part of the ratio you need.
Use fractions to find vectors along a line.

If $AP:PB = 2:3$ , then $AP = ____ AB$ .

Problem Solving with Vectors

Combine vector operations to find unknown vectors.
Use parallel conditions to prove collinearity or find ratios.
Always simplify your final answer.

Given $OA = 2a, OC = c$ , and $AB = 3c$ , what is OB?

$2a + 3c$
$3a + 2c$
$2a - 3c$
$3c - 2a$

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