7.1-7.4: Transformations and Vectors Notes:

Transformations

What are they?

• Transformations are ways to move or change shapes on a coordinate grid. You’ll learn how to describe and perform four types of transformations:

  1. Reflection

  2. Rotation

  3. Enlargement

  4. Translation

• Each transformation changes the shape’s position, size, or orientation, but not its basic structure.

Types of Transformations

1. Reflection

• A shape is flipped over a mirror line (e.g. x-axis, y-axis, or a vertical/horizontal line like x = 2).

• The shape stays the same size and orientation, but its position changes.

2. Rotation

• A shape is turned around a fixed point (usually the origin) by a given angle (90°, 180°, 270°).

• You must specify:

  • The center of rotation

  • The angle

  • The direction (clockwise or anticlockwise)

• In the exam, you can do this by tracing the shape on a tracing paper and then rotating it

3. Enlargement

• A shape is resized from a center of enlargement using a scale factor:

  • Scale factor > 1 → shape gets bigger

  • Scale factor < 1 → shape gets smaller

• The shape’s angles stay the same, but side lengths change proportionally.

4. Translation

• A shape is slid from one position to another using a vector.

• Vector is written as: ( x y ) in vertical orientation where:

  • x = movement left/right and y = movement up/down

Vectors

What Are Vectors?

• Vectors show movement from one point to another.

• They have both direction and magnitude (size).

• Written in column form where x is the horizontal movement and y is the vertical movement

• Notation: either the vector with an arrow on top OR the vector in bold

Example:

• A translation of 3 units right and 2 units up is:

Vector Operations

1. Add or subtract vectors by adding/subtracting components:

2. Multiply a vector by a scalar:

Examples:

• If a(2,5) and b(-1,3) then:

Magnitude of a Vector

• Formula: |v| = √(x² + y²)

|v| = √(6² + 8²) = √(36 + 64) = √100 = 10

Vector Geometry:

Using Vectors to Solve Geometric Problems

1. Position Vectors

• A position vector describes the location of a point relative to the origin or another point.

• Written as:

• where A is the starting point and B is the ending point. 

Example:

• If A = (2, 3) and B = (5, 7), then:

Vector Addition and Subtraction

• You can add vectors to find a new position or subtract to find displacement.

• ⃗AC = ⃗AB + ⃗BC

• ⃗BA = –⃗AB (reversing direction)

Example:

Midpoints Using Vectors

• The midpoint M of AB is found by averaging coordinates: OM = ½(OA + OB) where O is the origin

• In other words, add the coordinates of A and B then half them

Example:

Vector Proofs in Geometry

• Use vectors to prove:

  • Points are collinear (vectors are scalar multiples)

  • Triangles are isosceles or equilateral (equal magnitude)

  • Parallelograms (opposite sides equal and parallel)

1. To prove triangle ABC is isosceles:

• You calculate the distance of the 2 legs (e.g.: AB and AC). If they're equal, it's an isosceles

2. Showing that vectors are parallel

• Two vectors are parallel if they have the same direction.

• This means one vector is a scaled version of the other.

• You can check this by dividing the components — if both x and y parts scale by the same number, they’re parallel.

• E.g.: if a (2,3) and b (4, 6) then they're parallel since b = 2 x a (note: vectors should ALWAYS be in vertical orientation)

3. Showing That Three Points Are Collinear

• Points are collinear if they lie on the same straight line.

• In vector terms, this means the vectors between the points are parallel.

• You find the vectors between the points and check if they’re scalar multiples.

• Example:

  • A(1, 2), B(3, 6), C(5, 10)

⃗AB and ⃗BC are equal → Points are collinear ✅

4. Solving Vector Problems Involving Ratio and Similarity

1. Ratio Problems

• • When a point divides a line in a ratio (e.g. 2:1), you use vector fractions.

  • The position vector of the point is a weighted average of the endpoints.

  • Example:

    • Point P divides AB in ratio 2:1 AP = ⅔ × AB

2. Similarity Problems

• • Similar shapes have the same angles and proportional sides.

  • Vectors in similar shapes scale by the same factor.

  • You use scale factors to find missing sides or positions.

  • Example:

    • If triangle A′B′C′ is similar to triangle ABC with scale factor ½, then: A′B′ = ½ × AB

More details in the exam-style questions.

NOTES DONE BY FARIDA SABET

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