Vectors

This page covers vector notation and operations.

Key points:

• Vectors can be represented as column vectors or with letters (usually underlined)
• Vector addition and subtraction can be visualized geometrically
• Vector equations can be used to prove points lie on a straight line

Definition: A vector is a quantity with both magnitude and direction

Example: In the equation OB = a, AB = 2b, BD = a-b, vectors are used to describe positions and movements between points

Highlight: To show points lie on a straight line, prove that one vector is a scalar multiple of another

Vector Problems

This page demonstrates how to solve more complex vector problems.

Key points:

• Use parallel sides of shapes to relate vectors
• Break down complex paths into simpler vector components
• Ratios can be used to find positions along vectors

Example: In a parallelogram ABCD, if DE:EC = 8:1, vector DE can be expressed as 8/9 of vector DC

Highlight: When solving vector problems, break down the solution into steps: identify known vectors, use relationships between parallel sides, and combine vector components.

Two-Way Tables

Two-way tables organize data into categories across two variables.

Key points:

• Fill in missing information by adding and subtracting known values
• Ensure row and column totals are consistent

Example: In a survey of 400 people (200 men, 200 women) about handedness:

• 63 people are left-handed
• 167 women are right-handed

Complete the two-way table using this information.

Highlight: When completing two-way tables, work systematically to fill in missing values, using row and column totals as checks.

Similarity

This page introduces the concept of similarity in geometry.

Key points:

• Similar shapes have the same angles and proportional sides
• Area scales with the square of the linear scale factor
• Volume scales with the cube of the linear scale factor

Definition: Similar shapes have the same shape but may be different sizes

Highlight: In similar shapes, corresponding angles are equal and corresponding sides are in proportion.

Simultaneous Equations

This page covers methods for solving simultaneous equations.

Key points:

• Rearrange equations into the form ax + by = c
• Match coefficients of one variable
• Add or subtract equations to eliminate one variable
• Solve for the remaining variable and substitute back

Example: Solve the simultaneous equations: 2x + 4y = 6 4x + 3y = -3

Highlight: When solving simultaneous equations, the goal is to eliminate one variable by adding or subtracting the equations.

Graphical Inequalities

This page explains how to represent inequalities graphically.

Key points:

• Convert inequalities to equations for graphing
• Use solid lines for ≥ or ≤, dotted lines for > or <
• Shade the region that satisfies all inequalities

Example: Graph the inequalities: x + y < 5 y ≤ x + 2 y > 1

Highlight: The solution region for a system of inequalities is the area where all individual inequalities are satisfied simultaneously.

Trigonometric Graphs

This page covers the graphs of sine, cosine, and tangent functions.

Key points:

• Sine and cosine have a period of 360°
• Tangent has a period of 180°
• Key features include intercepts, peaks, and asymptotes

Vocabulary: Period - The interval after which a trigonometric function repeats

Example: Estimate the solutions for sin x = 0.7 between 0° and 360°

Highlight: Understanding the shapes and key features of trigonometric graphs is crucial for solving trigonometric equations graphically.

Sine Rule

This page explains the sine rule and its applications.

Key points:

• The sine rule relates sides and angles in any triangle
• It can be used to find unknown sides or angles
• Be cautious with the ambiguous case for angles

Definition: Sine Rule: a/sin A = b/sin B = c/sin C, where lowercase letters represent sides and uppercase letters represent opposite angles

Example: In triangle ABC, if a = 7m, A = 53°, and B = 83°, find the length of side b

Highlight: The sine rule is particularly useful when you know two angles and one side, or two sides and an angle opposite one of them.

Cosine Rule

This page covers the cosine rule and its applications.

Key points:

• The cosine rule relates all three sides and one angle in a triangle
• It can be used when the sine rule is not applicable
• There are separate formulas for finding a side and finding an angle

Definition: Cosine Rule for sides: a² = b² + c² - 2bc cos A Cosine Rule for angles: cos A = $b² + c² - a²$ / (2bc)

Example: In triangle ABC, if a = 7m, b = 8m, and C = 83°, find the length of side c

Highlight: The cosine rule is particularly useful in SAS (Side-Angle-Side) situations where you know two sides and the included angle.

Scatter Graphs

This page discusses scatter graphs and correlation.

Key points:

• Scatter graphs show relationships between two variables
• Correlation can be positive, negative, or none
• The line of best fit helps estimate values and identify trends

Vocabulary: Correlation - The strength and direction of the relationship between two variables

Example: A scatter graph showing the relationship between temperature and number of visitors to a beach

Highlight: Outliers are points that don't fit the general trend and should be considered carefully when interpreting scatter graphs.