CIE AMath Summary 2

Chapter 9: TRIGONOMETRIC FUNCTIONS

Chapter 9: TRIGONOMETRIC FUNCTIONS

Introduction

Trigonometric functions describe periodic phenomena such as waves, oscillations, and circular motion. This summary covers periodic behaviour, the sine, cosine, and tangent functions, trigonometric equations, identities, and solving equations in quadratic form.

Periodic Behaviour

Definition

A function $f(x)$ is periodic if there exists a positive constant $P$ (the period) such that:

$ f(x+P)=f(x) \quad \text{for all } x \text{ in the domain} $

The smallest such $P$ is the fundamental period.

Properties of Periodic Functions

• The graph repeats every $P$ units.
• If $f$ has period $P$, then $f(kx)$ has period $\frac{P}{|k|}$.
• The sum of periodic functions may not be periodic unless periods are commensurable.

Periods of Trigonometric Functions

Function	Principal Period
$\sin x$, $\cos x$, $\csc x$, $\sec x$	$2\pi$
$\tan x$, $\cot x$	$\pi$

Examples

1. $\sin(2x)$ has period $ \frac{2\pi}{2}=\pi $
2. $ \cos\left(\frac{x}{3}\right) $ has period $ \frac{2\pi}{1/3}=6\pi $
3. $ \tan(4x) $ has period $ \frac{\pi}{4} $
4. The function $ f(x)=\sin x+\sin(2x) $ is periodic with period $2\pi$ (least common multiple of $2\pi$ and $\pi$).

The Sine Function

Definition

$ y=\sin x $ is defined as the $y$-coordinate of the point on the unit circle at angle $x$ radians.

Key Properties

• Domain: $\mathbb{R}$ (all real numbers)
• Range: $[-1,1]$
• Period: $2\pi$
• Amplitude: $1$
• Odd function: $\sin(-x)=-\sin x$
• Zeros: $x=n\pi$ where $n\in\mathbb{Z}$
• Maximum: $1$ at $x=\frac{\pi}{2}+2n\pi$
• Minimum: $-1$ at $x=\frac{3\pi}{2}+2n\pi$

Graph of $y=\sin x$

Click to view the Graph

Transformed Sine Function

For $ y=a\sin(bx-c)+d $ :

• Amplitude: $|a|$
• Period: $\frac{2\pi}{|b|}$
• Phase shift: $\frac{c}{b}$ (to the right if $c > 0$)
• Vertical shift: $d$

Examples

1. Find the amplitude, period, and phase shift of $ y=3\sin(2x-\pi) $.
   Amplitude $=3$, Period $ \frac{2\pi}{2}=\pi $, Phase shift $ \frac{\pi}{2} $ to the right.
2. Sketch $ y=\sin\left(x-\frac{\pi}{2}\right) $.
   This is $\sin x$ shifted right by $ \frac{\pi}{2} $ (equivalent to $-\cos x$).
3. Evaluate $ \sin\frac{5\pi}{6} $.
   $ \frac{5\pi}{6}=150^\circ $, $ \sin\frac{5\pi}{6} = \sin\frac{\pi}{6} = \frac{1}{2} $

The Cosine Function

Definition

$y=\cos x$ is defined as the $x$-coordinate of the point on the unit circle at angle $x$ radians.

Key Properties

• Domain: $\mathbb{R}$
• Range: $[-1,1]$
• Period: $2\pi$
• Amplitude: $1$
• Even function: $\cos(-x)=\cos x$
• Zeros: $x=\frac{\pi}{2}+n\pi$
• Maximum: $1$ at $x=2n\pi$
• Minimum: $-1$ at $x=\pi+2n\pi$

Graph of $y=\cos x$

Click to view the Graph

Relationship Between Sine and Cosine

$$ \sin x=\cos\left(x-\frac{\pi}{2}\right), \qquad \cos x=\sin\left(x+\frac{\pi}{2}\right) $$

Examples

1. Find the zeros of $y=2\cos(3x)$ on $[0,2\pi)$.

   $\cos(3x)=0 \implies 3x=\frac{\pi}{2}+n\pi \implies x=\frac{\pi}{6}+\frac{n\pi}{3}$

   For $n=0,1,2,3,4,5$: $$ x=\frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} $$

2. Evaluate $\cos\frac{4\pi}{3}$.

   $\frac{4\pi}{3}=240^\circ$, reference angle $\frac{\pi}{3}$ (Quadrant III, cosine negative).

   $$ \cos\frac{4\pi}{3} = -\cos\frac{\pi}{3} = -\frac12 $$

3. Find the maximum and minimum of $y=5-3\cos x$.

   $$ \cos x\in[-1,1] $$ $$ 5-3\cos x \in [5-3(1),\,5-3(-1)] = [2,8] $$

   Maximum $=8$, Minimum $=2$.

The Tangent Function

Definition

$$ \tan x=\frac{\sin x}{\cos x} $$

Key Properties

• Domain: $x\neq \frac{\pi}{2}+n\pi$ (where $\cos x=0$)
• Range: $\mathbb{R}$
• Period: $\pi$
• Odd function: $\tan(-x)=-\tan x$
• Zeros: $x=n\pi$
• Vertical asymptotes: $x=\frac{\pi}{2}+n\pi$
• Increasing on each interval $\left(-\frac{\pi}{2}+n\pi,\frac{\pi}{2}+n\pi\right)$

Graph of $y=\tan x$

Click to view the Graph

Transformed Tangent Function

For $y=a\tan(bx-c)+d$:

• Period: $\frac{\pi}{|b|}$
• Phase shift: $\frac{c}{b}$
• Vertical shift: $d$
• Asymptotes at $bx-c=\frac{\pi}{2}+n\pi$

Examples

1. Find the period and asymptotes of $y=\tan(2x)$.

   Period: $$ \frac{\pi}{2} $$

   Asymptotes: $$ 2x=\frac{\pi}{2}+n\pi $$ $$ x=\frac{\pi}{4}+\frac{n\pi}{2} $$

2. Evaluate $\tan\frac{3\pi}{4}$.

   $\frac{3\pi}{4}=135^\circ$ (Quadrant II, tangent negative).

   Reference angle: $$ \frac{\pi}{4} $$ $$ \tan\frac{3\pi}{4} = -\tan\frac{\pi}{4} = -1 $$

3. Solve $\tan x=\sqrt3$ for $x\in[0,2\pi)$.

   Reference angle: $$ \frac{\pi}{3} $$

   Tangent is positive in Quadrants I and III:

   $$ x=\frac{\pi}{3}, \qquad x=\frac{4\pi}{3} $$

Trigonometric Equations

General Solution Formulas

$$ \sin\theta=\sin\alpha \implies \theta=\alpha+2n\pi \quad\text{or}\quad \theta=\pi-\alpha+2n\pi $$

$$ \cos\theta=\cos\alpha \implies \theta=\pm\alpha+2n\pi $$

$$ \tan\theta=\tan\alpha \implies \theta=\alpha+n\pi $$

Solving Strategy

1. Isolate the trigonometric function.
2. Find the reference angle (principal value).
3. Determine all quadrants where the function has the given sign.
4. Write the general solution using periodicity.
5. Find required solutions in the given interval.

Examples

1. Solve $\sin x=\frac{\sqrt3}{2}$ for $x\in[0,2\pi)$.

   Reference angle: $$ \frac{\pi}{3} $$

   Sine is positive in Quadrants I and II:

   $$ x=\frac{\pi}{3}, \qquad x=\frac{2\pi}{3} $$

2. Solve $\cos x=-\frac12$ for $x\in\mathbb{R}$.

   $$ x=\frac{2\pi}{3}+2n\pi $$ or $$ x=\frac{4\pi}{3}+2n\pi $$

3. Solve $\tan x=-1$ for $x\in[0,2\pi)$.

   $$ x=\frac{3\pi}{4}, \qquad x=\frac{7\pi}{4} $$

4. Solve $2\sin x+1=0$ for $x\in[0,2\pi)$.

   $$ \sin x=-\frac12 $$

   $$ x=\frac{7\pi}{6}, \qquad x=\frac{11\pi}{6} $$

Trigonometric Relationships (Identities)

Fundamental Identities

$$ \sin^2 x + \cos^2 x = 1 $$ $$ 1 + \tan^2 x = \sec^2 x $$ $$ 1 + \cot^2 x = \csc^2 x $$

Reciprocal Identities

$$ \csc x = \frac{1}{\sin x}, \quad \sec x = \frac{1}{\cos x}, \quad \cot x = \frac{1}{\tan x} $$

Quotient Identities

$$ \tan x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x} $$

Even-Odd Identities

$$ \sin(-x) = -\sin x \quad \text{(odd)} $$ $$ \cos(-x) = \cos x \quad \text{(even)} $$ $$ \tan(-x) = -\tan x \quad \text{(odd)} $$

Co-function Identities

$$ \sin\left(\frac{\pi}{2}-x\right)=\cos x,\quad \cos\left(\frac{\pi}{2}-x\right)=\sin x $$ $$ \tan\left(\frac{\pi}{2}-x\right)=\cot x $$

Addition and Subtraction Formulas

$$ \sin(A+B)=\sin A\cos B+\cos A\sin B $$ $$ \sin(A-B)=\sin A\cos B-\cos A\sin B $$ $$ \cos(A+B)=\cos A\cos B-\sin A\sin B $$ $$ \cos(A-B)=\cos A\cos B+\sin A\sin B $$ $$ \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B} $$ $$ \tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B} $$

Double Angle Formulas

$$ \sin(2x)=2\sin x\cos x $$ $$ \cos(2x)=\cos^2 x-\sin^2 x =2\cos^2 x-1 =1-2\sin^2 x $$ $$ \tan(2x)=\frac{2\tan x}{1-\tan^2 x} $$

Examples

1. Simplify $ \frac{\sin^2 x}{1-\cos x} $.

   $ \frac{\sin^2 x}{1-\cos x} = \frac{1-\cos^2 x}{1-\cos x} = \frac{(1-\cos x)(1+\cos x)}{1-\cos x} = 1+\cos x $

2. Prove $ \tan x \sin x + \cos x = \sec x $.

   LHS $ = \frac{\sin x}{\cos x}\cdot\sin x+\cos x = \frac{\sin^2 x}{\cos x}+\cos x = \frac{\sin^2 x+\cos^2 x}{\cos x} = \frac{1}{\cos x} = \sec x $

3. Find $ \cos 75^\circ $ exactly.

   $ \cos 75^\circ = \cos(45^\circ+30^\circ) = \cos45^\circ\cos30^\circ - \sin45^\circ\sin30^\circ $

   $ = \frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2} - \frac{\sqrt2}{2}\cdot\frac12 = \frac{\sqrt6-\sqrt2}{4} $

4. Simplify $ \sin(2x)\cos x+\cos(2x)\sin x $.

   Using the sine addition formula:

   $ \sin(2x)\cos x+\cos(2x)\sin x = \sin(2x+x) = \sin(3x) $

Trigonometric Equations in Quadratic Form

Method

Equations of the form $a\sin^2 x+b\sin x+c=0$ (or with $\cos x$ or $\tan x$) are solved by:

1. Substitute $u=\sin x$ (or $\cos x$, $\tan x$).
2. Solve the quadratic $au^2+bu+c=0$.
3. Back-substitute and solve for $x$.
4. Check for extraneous solutions (especially when squaring or using $\tan$).

Examples

1. Solve $2\sin^2 x-\sin x-1=0$ for $x\in[0,2\pi)$.

   Let $u=\sin x$:

   $ 2u^2-u-1=0 $

   $ (2u+1)(u-1)=0 \implies u=-\frac12 \text{ or } u=1 $

   $ \sin x=-\frac12 \Rightarrow x=\frac{7\pi}{6},\frac{11\pi}{6} $

   $ \sin x=1 \Rightarrow x=\frac{\pi}{2} $

   Solutions: $ x=\frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6} $

2. Solve $2\cos^2 x-3\cos x+1=0$ for $x\in[0,2\pi)$.

   Let $u=\cos x$:

   $ 2u^2-3u+1=0 $

   $ (2u-1)(u-1)=0 \implies u=\frac12 \text{ or } u=1 $

   $ \cos x=\frac12 \Rightarrow x=\frac{\pi}{3},\frac{5\pi}{3} $

   $ \cos x=1 \Rightarrow x=0 $

   Solutions: $ x=0, \frac{\pi}{3}, \frac{5\pi}{3} $

3. Solve $\tan^2 x-3=0$ for $x\in[0,2\pi)$.

   $ \tan^2 x=3 \Rightarrow \tan x=\pm\sqrt3 $

   $ \tan x=\sqrt3 \Rightarrow x=\frac{\pi}{3}, \frac{4\pi}{3} $

   $ \tan x=-\sqrt3 \Rightarrow x=\frac{2\pi}{3}, \frac{5\pi}{3} $

   Solutions: $ x= \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} $

4. Solve $\sin^2 x-\cos^2 x=0$ for $x\in[0,2\pi)$.

   Using $ \cos^2 x=1-\sin^2 x $

   $ \sin^2 x-(1-\sin^2 x)=0 $

   $ 2\sin^2 x-1=0 \Rightarrow \sin^2 x=\frac12 \Rightarrow \sin x=\pm\frac{\sqrt2}{2} $

   Solutions: $ x= \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} $

5. Solve $3\sec^2 x-4=0$ for $x\in[0,2\pi)$.

   $ \sec^2 x=\frac43 \Rightarrow \cos^2 x=\frac34 \Rightarrow \cos x=\pm\frac{\sqrt3}{2} $

   $ \cos x=\frac{\sqrt3}{2} \Rightarrow x=\frac{\pi}{6}, \frac{11\pi}{6} $

   $ \cos x=-\frac{\sqrt3}{2} \Rightarrow x=\frac{5\pi}{6}, \frac{7\pi}{6} $

   Solutions: $ x= \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} $

Using Identities to Create Quadratic Form

Sometimes the equation needs to be rewritten using identities before substitution.

Example: Using $ \cos 2x $ Identity

Solve $ \cos 2x+\cos x=0 $ for $x\in[0,2\pi)$.

Using $ \cos 2x=2\cos^2 x-1 $:

$$ 2\cos^2 x-1+\cos x=0 \implies 2\cos^2 x+\cos x-1=0 $$

Let $u=\cos x$:

$$ 2u^2+u-1=0 \implies (2u-1)(u+1)=0 $$ $$ u=\frac12 \quad\text{or}\quad u=-1 $$

$ \cos x=\frac12 \Rightarrow x=\frac{\pi}{3}, \frac{5\pi}{3} $

$ \cos x=-1 \Rightarrow x=\pi $

Solutions: $ x= \frac{\pi}{3}, \pi, \frac{5\pi}{3} $

Summary Table

Concept	Key Points
Periodicity	$\sin$, $\cos$ period $2\pi$; $\tan$ period $\pi$
Sine	Domain $\mathbb{R}$, range $[-1,1]$, odd
Cosine	Domain $\mathbb{R}$, range $[-1,1]$, even
Tangent	Domain $x \neq \frac{\pi}{2}+n\pi$, range $\mathbb{R}$, odd
General solutions	$\sin\theta=\sin\alpha \Rightarrow \theta=\alpha+2n\pi$ or $\pi-\alpha+2n\pi$
	$\cos\theta=\cos\alpha \Rightarrow \theta=\pm\alpha+2n\pi$
	$\tan\theta=\tan\alpha \Rightarrow \theta=\alpha+n\pi$
Pythagorean identities	$\sin^2\theta+\cos^2\theta=1$, $1+\tan^2\theta=\sec^2\theta$
Double angle	$\sin2\theta=2\sin\theta\cos\theta$, $\cos2\theta=\cos^2\theta-\sin^2\theta$
Quadratic equations	Substitute $u=\sin x$ (or $\cos x$, $\tan x$), solve, back-substitute

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Chapter 10: COUNTING AND THE BINOMIAL EXPANSION

Chapter 10: COUNTING AND THE BINOMIAL EXPANSION

Introduction

How many different passwords can you create using 4 letters? How many ways can a committee of 3 people be chosen from a group of 10? How many possible outcomes are there when flipping a coin 5 times? These questions all belong to the branch of mathematics known as combinatorics — the art of counting without listing every possibility.

At first glance, counting might seem too simple to deserve serious study. However, when the number of possibilities grows into the millions or billions, systematic methods become essential. Moreover, the techniques we develop for counting lead directly to one of the most elegant and powerful results in algebra: the Binomial Theorem.

Why Counting Matters

Counting is not just about abstract numbers; it appears everywhere:

• Computer Science: Estimating the number of possible passwords, IP addresses, or algorithm outcomes.
• Probability: Calculating the chance of winning the lottery, drawing a specific card, or getting a certain number of heads in coin tosses.
• Genetics: Determining the number of possible genetic combinations from parents.
• Networks: Finding the number of possible routes or connections.
• Sports tournaments: Calculating the number of ways to arrange matches or choose teams.

Two Fundamental Principles

Most counting problems are built from two simple ideas:

1. The Product Principle (Multiplication Principle):
   If a task can be broken into a sequence of steps, and the number of ways to perform each step does not depend on previous choices, we multiply the number of options at each step.

   Example: Choosing a meal with 3 appetizers, 4 main courses, and 2 desserts gives $3 \times 4 \times 2 = 24$ possible meals.

2. The Sum Principle (Addition Principle):
   If a task can be done in one of several mutually exclusive ways, we add the number of options for each way.

   Example: Choosing a vowel (A, E, I, O, U) gives $5$ possibilities; choosing a consonant gives $21$; choosing either gives $5 + 21 = 26$ letters.

From Counting to the Binomial Expansion

Remarkably, the same numbers that count combinations — known as binomial coefficients $\binom{n}{r}$ — appear when we expand expressions like $(a+b)^n$. For example:

$$(a+b)^2 = a^2 + 2ab + b^2$$

$$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$

The coefficients $1,2,1$ and $1,3,3,1$ are exactly the numbers of ways to choose 0, 1, or 2 items from 2 items, and 0, 1, 2, or 3 items from 3 items. This is not a coincidence — it is the heart of the Binomial Theorem.

Pascal's Triangle

The binomial coefficients also form a famous triangular array called Pascal's Triangle, where each number is the sum of the two above it:

        1
      1   1
    1   2   1
  1   3   3   1
1   4   6   4   1
      ...

Pascal's Triangle appears in probability, algebra, combinatorics, and even in fractal geometry.

What You Will Learn

• A — The Product Principle — the foundation of counting.
• B — Counting paths in grids — a geometric application.
• C — Factorial notation — a compact way to handle large products.
• D — Permutations — ordered arrangements.
• E — Combinations — unordered selections.
• F — Binomial expansions — patterns in $(a+b)^n$.
• G — The Binomial Theorem — the general formula connecting counting and algebra.

A Bridge Between Two Worlds

Counting and the Binomial Expansion together form a beautiful bridge between discrete mathematics (counting finite sets) and algebra (expanding expressions). By the end of this topic, you will see that choosing a committee of 3 from 10 people and finding the coefficient of $a^7b^3$ in $(a+b)^{10}$ are actually the same problem, viewed from different angles.

Whether you are interested in probability, computer science, or pure mathematics, mastering these ideas will give you powerful tools for solving real-world and theoretical problems.

A. The Product Principle

• Definition: If a task can be broken into a sequence of $k$ steps, where the first step can be done in $n_1$ ways, the second in $n_2$ ways (regardless of the first), and so on, then the total number of ways to complete the task is $n_1 \times n_2 \times \cdots \times n_k$.
• Also known as the multiplication principle.

Example. A restaurant offers 3 appetizers, 4 main courses, and 2 desserts. How many different 3-course meals can you order?

$$3 \times 4 \times 2 = 24 \text{ meals.}$$

Example. How many 3-letter codes can be formed from the letters A, B, C, D if repetition is allowed?

$$4 \times 4 \times 4 = 4^3 = 64.$$

B. Counting Paths

• Counting the number of routes in a grid from one corner to another using only moves right (R) and up (U).
• A path from $(0,0)$ to $(m,n)$ requires exactly $m$ R's and $n$ U's. The total number of paths is:

$$\binom{m+n}{m} = \binom{m+n}{n}.$$

Example. How many shortest paths from $(0,0)$ to $(3,2)$ moving only right or up?

$$\text{Total steps} = 3+2 = 5,\quad \binom{5}{3} = \binom{5}{2} = 10 \text{ paths.}$$

C. Factorial Notation

• Definition: For a non-negative integer $n$, $n!$ (read “n factorial”) is:

$$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1.$$

• Special values: $0! = 1$, $1! = 1$.

Example. Compute:

$$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120,\quad \frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!} = 42.$$

Example. Simplify $\frac{n!}{(n-2)!}$ for $n \geq 2$:

$$\frac{n \times (n-1) \times (n-2)!}{(n-2)!} = n(n-1).$$

D. Permutations

• A permutation is an ordered arrangement of distinct objects.
• Number of permutations of $n$ distinct objects taken $r$ at a time:

$$P(n,r)=\frac{n!}{(n-r)!}.$$

• If all $n$ objects are arranged: $n!$.

Example. How many ways to arrange 5 books on a shelf?

$$5! = 120.$$

Example. How many 4-letter codes can be formed from the letters A, B, C, D, E, F without repetition?

$$P(6,4)=\frac{6!}{2!}=6\times5\times4\times3=360.$$

E. Combinations

• A combination is an unordered selection of distinct objects.
• Number of combinations of $n$ distinct objects taken $r$ at a time:

$$\binom{n}{r}=C(n,r)=\frac{n!}{r!(n-r)!}.$$

• Properties: $\binom{n}{r}=\binom{n}{n-r}$, $\binom{n}{0}=\binom{n}{n}=1$.

Example. From a group of 10 people, how many ways to choose a committee of 3?

$$\binom{10}{3}=\frac{10!}{3!7!}=\frac{10\times9\times8}{3\times2\times1}=120.$$

Example. A pizza shop offers 8 toppings. How many 3-topping pizzas?

$$\binom{8}{3}=56.$$

F. Binomial Expansions

• Binomial expansion expands $(a+b)^n$ using binomial coefficients.
• Pattern: $(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$.
• The coefficients are the entries of Pascal's Triangle.

Example. Expand $(x+y)^3$:

$$\binom{3}{0}x^3+\binom{3}{1}x^2y+\binom{3}{2}xy^2+\binom{3}{3}y^3=x^3+3x^2y+3xy^2+y^3.$$

Example. Find the term in $x^5$ in $(2x-3)^7$.

The general term: $\binom{7}{k}(2x)^{7-k}(-3)^k$. We need $7-k=5 \Rightarrow k=2$.

Term = $\binom{7}{2}(2x)^5(-3)^2 = 21 \times 32x^5 \times 9 = 6048x^5$.

G. The Binomial Theorem

• Theorem: For any real numbers $a$, $b$ and non-negative integer $n$:

$$ (a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k. $$

• Equivalent form: $\sum_{k=0}^{n}\binom{n}{k}a^k b^{n-k}$.

Example. Use the Binomial Theorem to expand $(1+2x)^4$.

$$ (1)^4+\binom{4}{1}(1)^3(2x)+\binom{4}{2}(1)^2(2x)^2+\binom{4}{3}(1)(2x)^3+\binom{4}{4}(2x)^4 $$

$$ =1+4(2x)+6(4x^2)+4(8x^3)+16x^4 =1+8x+24x^2+32x^3+16x^4. $$

Example. Find the coefficient of $x^3$ in $(2-x)^5$.

General term: $\binom{5}{k}(2)^{5-k}(-x)^k=\binom{5}{k}2^{5-k}(-1)^k x^k$.

Set $k=3$: $\binom{5}{3}2^2(-1)^3=10\times4\times(-1)=-40$.

Coefficient: $-40$.

Summary Table

Concept	Formula
Product Principle	$n_1 \times n_2 \times \cdots \times n_k$
Permutations	$P(n,r)=\frac{n!}{(n-r)!}$
Combinations	$\binom{n}{r}=\frac{n!}{r!(n-r)!}$
Binomial Theorem	$(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$

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Chapter 11: VECTORS

Chapter 11: VECTORS

Introduction

In many areas of mathematics, physics, and engineering, we encounter quantities that require more than just a number to be fully described. For example, telling someone that a car is traveling at 60 km/h does not tell them where it is going; the direction is essential. Similarly, describing a force of 10 Newtons is incomplete without specifying whether it is pushing up, down, left, or right. This leads us to a fundamental distinction between two types of quantities:

A. Vectors and Scalars

• Scalar: A quantity with only magnitude (size). Examples: mass, temperature, time, speed.
• Vector: A quantity with both magnitude and direction. Examples: displacement, velocity, acceleration, force.
• Notation: $\vec{v}$, $\mathbf{v}$, or $\overrightarrow{AB}$.

Example. Identify each as scalar or vector:

1. 5 kg (scalar)
2. 20 m/s north (vector)
3. 100 J of energy (scalar)
4. Force of 10 N downwards (vector)

B. The Magnitude of a Vector

• The magnitude (length) of a vector $\vec{v}=\begin{pmatrix}x\\y\end{pmatrix}$ is:

$$|\vec{v}|=\sqrt{x^2+y^2}.$$

• For $\overrightarrow{AB}$ where $A(x_1,y_1)$ and $B(x_2,y_2)$:

$$|\overrightarrow{AB}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.$$

• A unit vector has magnitude 1: $\hat{v}=\frac{\vec{v}}{|\vec{v}|}$.

Example. Find the magnitude of $\vec{v}=\begin{pmatrix}3\\-4\end{pmatrix}$.

$$ |\vec{v}|=\sqrt{3^2+(-4)^2} =\sqrt{9+16} =\sqrt{25} =5. $$

Example. Given $A(1,2)$ and $B(4,6)$, find $|\overrightarrow{AB}|$.

$$ \overrightarrow{AB} = \begin{pmatrix} 4-1\\ 6-2 \end{pmatrix} = \begin{pmatrix} 3\\ 4 \end{pmatrix}, \quad |\overrightarrow{AB}| = \sqrt{3^2+4^2} =5. $$

C. Operations with Plane Vectors

• Addition: $\vec{u}+\vec{v}=\begin{pmatrix}u_x+v_x\\u_y+v_y\end{pmatrix}$ (Triangle/Parallelogram law).
• Scalar multiplication: $k\vec{v}=\begin{pmatrix}kx\\ky\end{pmatrix}$.
• Negative: $-\vec{v}=\begin{pmatrix}-x\\-y\end{pmatrix}$.
• Subtraction: $\vec{u}-\vec{v}=\vec{u}+(-\vec{v})$.

Example. Let $\vec{u}=\begin{pmatrix}2\\3\end{pmatrix}$, $\vec{v}=\begin{pmatrix}-1\\4\end{pmatrix}$.

$$ \vec{u}+\vec{v} = \begin{pmatrix} 2+(-1)\\ 3+4 \end{pmatrix} = \begin{pmatrix} 1\\ 7 \end{pmatrix}, \quad 3\vec{u} = \begin{pmatrix} 6\\ 9 \end{pmatrix}, \quad \vec{u}-\vec{v} = \begin{pmatrix} 3\\ -1 \end{pmatrix}. $$

D. The Vector Between Two Points

• If $A(x_1,y_1)$ and $B(x_2,y_2)$, then:

$$ \overrightarrow{AB} = \begin{pmatrix} x_2-x_1\\ y_2-y_1 \end{pmatrix}. $$

• Position vector of a point: $\vec{r}_A=\overrightarrow{OA}$ where $O$ is origin.
• $\overrightarrow{AB}=\vec{r}_B-\vec{r}_A$.

Example. If $A(3,-2)$ and $B(-1,5)$, find $\overrightarrow{AB}$.

$$ \overrightarrow{AB} = \begin{pmatrix} -1-3\\ 5-(-2) \end{pmatrix} = \begin{pmatrix} -4\\ 7 \end{pmatrix}. $$

Example. Given $\vec{r}_A=\begin{pmatrix}2\\1\end{pmatrix}$, $\vec{r}_B=\begin{pmatrix}5\\4\end{pmatrix}$, find $\overrightarrow{AB}$.

$$ \overrightarrow{AB} = \vec{r}_B-\vec{r}_A = \begin{pmatrix} 5-2\\ 4-1 \end{pmatrix} = \begin{pmatrix} 3\\ 3 \end{pmatrix}. $$

E. Parallelism

• Two non-zero vectors $\vec{u}$ and $\vec{v}$ are parallel if one is a scalar multiple of the other:

$$ \vec{u}=k\vec{v} \quad \text{for some real } k \neq 0. $$

• If $k>0$: same direction; if $k\lt0$: opposite direction.
• For collinearity: points $A,B,C$ are collinear if $\overrightarrow{AB}\parallel\overrightarrow{AC}$.

Example. Show $\vec{u}=\begin{pmatrix}4\\-6\end{pmatrix}$ is parallel to $\vec{v}=\begin{pmatrix}-2\\3\end{pmatrix}$.

$$ \vec{u} = -2 \begin{pmatrix} -2\\ 3 \end{pmatrix} = -2\vec{v} \quad\Rightarrow\quad \text{parallel (opposite direction)}. $$

Example. Check if $A(1,2)$, $B(3,5)$, $C(7,11)$ are collinear.

$$ \overrightarrow{AB} = \begin{pmatrix} 2\\ 3 \end{pmatrix}, \quad \overrightarrow{AC} = \begin{pmatrix} 6\\ 9 \end{pmatrix} = 3\overrightarrow{AB} \quad\Rightarrow\quad \text{collinear}. $$

F. Problems Involving Vector Operations

• Use vector addition, subtraction, and scalar multiplication to solve geometric problems.
• Midpoint: $M$ of $AB$ has position vector $\frac{\vec{r}_A+\vec{r}_B}{2}$.
• Section formula: Point dividing $AB$ in ratio $\lambda:\mu$ has position vector $\frac{\mu\vec{r}_A+\lambda\vec{r}_B}{\lambda+\mu}$.

Example. Find the midpoint of $A(2,3)$ and $B(6,7)$.

$$ \vec{r}_M = \frac12 \left( \begin{pmatrix} 2\\ 3 \end{pmatrix} + \begin{pmatrix} 6\\ 7 \end{pmatrix} \right) = \begin{pmatrix} 4\\ 5 \end{pmatrix}. $$

Example. Given $\vec{a}=\begin{pmatrix}1\\2\end{pmatrix}$, $\vec{b}=\begin{pmatrix}4\\6\end{pmatrix}$, find the vector from $A$ to the midpoint of $OB$, where $O$ is origin.

Midpoint of $OB$:

$$ \frac{\vec{b}}{2} = \begin{pmatrix} 2\\ 3 \end{pmatrix}. $$

Then

$$ \overrightarrow{A\text{mid}} = \begin{pmatrix} 2-1\\ 3-2 \end{pmatrix} = \begin{pmatrix} 1\\ 1 \end{pmatrix}. $$

G. Lines

• Vector form: $\vec{r}=\vec{a}+t\vec{d}$, where $\vec{a}$ is a point on the line, $\vec{d}$ is a direction vector, $t\in\mathbb{R}$.
• Parametric form: $x=a_1+td_1,\; y=a_2+td_2$.
• Cartesian form: Eliminate $t$: $\frac{x-a_1}{d_1}=\frac{y-a_2}{d_2}$ (if $d_1,d_2\neq0$).

Example. Find vector equation of line through $A(1,2)$ parallel to $\vec{d}=\begin{pmatrix}3\\-1\end{pmatrix}$.

$$ \vec{r} = \begin{pmatrix} 1\\ 2 \end{pmatrix} + t \begin{pmatrix} 3\\ -1 \end{pmatrix}. $$

Example. Convert to Cartesian: $x=1+3t,\; y=2-t$.

$$ t=\frac{x-1}{3}, \qquad t=2-y. $$

Therefore

$$ \frac{x-1}{3} = 2-y \Rightarrow x-1=6-3y \Rightarrow x+3y=7. $$

H. Constant Velocity Problems

• Position vector at time $t$: $\vec{r}(t)=\vec{r}_0+\vec{v}t$, where $\vec{r}_0$ is initial position and $\vec{v}$ is constant velocity.
• Velocity: $\vec{v}=\frac{\text{displacement}}{\text{time}}$.
• Relative velocity: $\vec{v}_{A/B}=\vec{v}_A-\vec{v}_B$.
• Collision/intersection: Solve $\vec{r}_A(t)=\vec{r}_B(s)$ for $t,s$.

Example. A particle starts at $(2,3)$ with velocity $\begin{pmatrix}4\\-1\end{pmatrix}$. Find position at $t=5$.

$$ \vec{r}(5) = \begin{pmatrix} 2\\ 3 \end{pmatrix} + 5 \begin{pmatrix} 4\\ -1 \end{pmatrix} = \begin{pmatrix} 22\\ -2 \end{pmatrix}. $$

Example. Two ships: Ship A at $(0,0)$ moving east at 20 km/h, Ship B at $(10,0)$ moving north at 15 km/h. Find relative velocity of A relative to B.

$$ \vec{v}_A= \begin{pmatrix} 20\\ 0 \end{pmatrix}, \quad \vec{v}_B= \begin{pmatrix} 0\\ 15 \end{pmatrix}, \quad \vec{v}_{A/B} = \begin{pmatrix} 20\\ -15 \end{pmatrix} \text{ km/h}. $$

$$ \sqrt{20^2+(-15)^2} = \sqrt{400+225} = \sqrt{625} = 25 \text{ km/h}. $$

Example. Do they collide?

$$ \vec{r}_A(t) = \begin{pmatrix} 20t\\ 0 \end{pmatrix}, \qquad \vec{r}_B(s) = \begin{pmatrix} 10\\ 15s \end{pmatrix}. $$

Equate components:

$$ 20t=10 \Rightarrow t=0.5, \qquad 0=15s \Rightarrow s=0. $$

Different times $\Rightarrow$ no collision.

Summary Table

Concept	Formula
Magnitude	$|\vec{v}|=\sqrt{x^2+y^2}$
Vector between points	$\overrightarrow{AB}=\vec{r}_B-\vec{r}_A$
Parallelism	$\vec{u}=k\vec{v}$
Line (vector form)	$\vec{r}=\vec{a}+t\vec{d}$
Constant velocity	$\vec{r}(t)=\vec{r}_0+\vec{v}t$
Relative velocity	$\vec{v}_{A/B}=\vec{v}_A-\vec{v}_B$