Lesson Notes By Weeks and Term v4 - JHS 2

Lesson Video

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Performance objectives

• Define a vector and its components.
• Identify when two vectors are equal.
• Illustrate vector equality using diagrams.
• Solve problems involving vector equality.

Lesson summary

This lesson introduces the concept of vectors and focuses on what it means for two vectors to be equal. In our daily lives in Ghana, we are constantly dealing with quantities that have both size and direction. For example, when giving directions to a friend to find your house, you might say "Walk 100 metres towards the market." The "100 metres" is the size (magnitude), and "towards the market" is the direction. This is a vector! Understanding when two such "journeys" or "forces" are the same is important in many fields like physics (forces), navigation (sailing on the Volta Lake or flying from Accra to Tamale), and even computer animation.

Lesson notes

a. What is a Vector? (Recap)

First, let's remember the difference between a scalar and a vector. Scalar: A quantity that has only magnitude (size or amount). *Examples:* The price of a ball of kenkey (GH₵5), your age (14 years), the distance from Accra to Kumasi (250 km), the temperature (32°C). Vector: A quantity that has both magnitude and direction. *Examples:* A car travelling at 60 km/h due north, a force of 20 Newtons pushing a table to the right, a displacement of 3km south-west. b. Representing Vectors

We can represent vectors in two main ways: Graphically: As an arrow. The length of the arrow represents the magnitude. The direction the arrow points represents the direction of the vector. We can name a vector with a small letter (e.g., a, b) or by its start and end points (e.g., $\vec{AB}$, meaning the vector from point A to point B). Using Column Vectors: This is a very useful way to describe a vector's movement on a grid (Cartesian plane). A column vector is written as $\begin{pmatrix} x \\ y \end{pmatrix}$. The top number, x, tells us the horizontal movement (right is positive, left is negative). The bottom number, y, tells us the vertical movement (up is positive, down is negative).

Worked Example: Finding a Column Vector Find the column vector for the vector $\vec{PQ}$ starting at point P(1, 2) and ending at point Q(5, 5). Step 1: Find the horizontal movement (x). We move from x=1 to x=5. The change is $x_Q - x_P = 5 - 1 = 4$. (4 units to the right). Step 2: Find the vertical movement (y). We move from y=2 to y=5. The change is $y_Q - y_P = 5 - 2 = 3$. (3 units up). Step 3: Write it as a column vector. $\vec{PQ} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}$ c. The Core Concept: Vector Equality