Lesson Notes By Weeks and Term v4 - SHS 2
SPATIAL SENSE
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SPATIAL SENSE
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Subject: Additional Mathematics
Class: SHS 2
Term: 2nd Term
Week: 5
Grade code: 2.2.1.LI.3
Strand code: 2
Sub-strand code: 1
Content standard code: 2.2.1.CS.2
Indicator code: 2.2.1.LI.3
Theme: GEOMETRIC REASONING AND MEASUREMENT
Subtheme: SPATIAL SENSE
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Welcome, learners. In our previous lessons, we explored vectors as tools for representing quantities with both magnitude and direction. Today, we will see how truly powerful they are by using them to connect algebra and geometry. We will derive two of the most important rules in trigonometry—the Cosine and Sine Rules—using vector principles. We will also learn how to use vectors to calculate the area of a piece of land or any triangular shape without needing to measure its height directly. This skill is vital in fields like land surveying, architecture, and even in navigation for our fishermen at sea.
A. Prerequisite Knowledge (Quick Recap)
Before we begin, let's remember three key vector concepts: Triangle Law of Vector Addition: For a triangle PQR, `PQ` + `QR` = `PR`. This means we can express one side in terms of the others, for example, `QR` = `PR` - `PQ`. Magnitude of a Vector: If vector a = (x, y), its magnitude is |a| = √(x² + y²). The Dot (Scalar) Product: For two vectors a and b, their dot product is defined in two ways: Algebraically: If a = (x₁, y₁) and b = (x₂, y₂), then a · b = x₁x₂ + y₁y₂. Geometrically: a · b = |a| |b| cos θ, where θ is the angle between the vectors. A special case: a · a = |a| |a| cos 0° = |a|². B. Deriving the Cosine Rule using Vectors
This is a beautiful proof that shows the link between vector algebra and triangle geometry.
Consider a triangle PQR. Let the position vectors of the vertices be p, q, and r. Let the side vectors be: PR = r - p (let's call this vector q' for simplicity, with magnitude *q*) PQ = q - p (let's call this vector r' for simplicity, with magnitude *r*) QR = r - q (let's call this vector p' for simplicity, with magnitude *p*)