Vectors

Welcome to the world of Vectors! In this module of Maths Extension 1, we'll explore a powerful mathematical tool for representing quantities that have both size and direction.

Think about giving directions: "Walk 5 blocks." This tells you how far to walk (the magnitude), but not which way to walk (the direction). To give complete directions, you'd need to say something like "Walk 5 blocks north." This is where vectors come in.

A vector is a mathematical object that has both magnitude (size or length) and direction. Quantities that only have magnitude (like temperature, mass, or just regular numbers) are called scalars. So, 5 is a scalar, while "5 blocks north" is a vector.

We often represent vectors visually as arrows. The length of the arrow corresponds to the vector's magnitude, and the arrowhead indicates its direction.

Vector Algebra

This module will introduce you to vector algebra, which is a way of performing operations on vectors. We'll learn how to combine vectors using operations similar to addition and subtraction with scalars. These are called vector addition and vector subtraction.

Multiplying vectors is a bit more nuanced than multiplying scalars. It's not immediately obvious what "multiplying two directions" would mean. Instead of a simple multiplication, we'll explore different types of "products" involving vectors:

• Scalar Multiplication: Multiplying a vector by a scalar to construct a new vector
• Dot Product (Scalar Product): A way of multiplying two vectors that results in a scalar.

Applications of Vectors

Vectors are incredibly useful in many areas of mathematics, science, and engineering. In this module, we'll focus on two key applications:

• Geometric Proofs: Using vectors to prove geometric theorems and solve geometric problems. This can often provide elegant and concise solutions.
• Projectile Motion: Applying vector algebra to analyze the motion of objects through the air, such as a ball being thrown or a rocket being launched. This is a common and important application in physics.

Get ready to explore the exciting world of vectors!

• Describing Vectors
• Defining Vector Addition
• The zero vector and vector inverses
• Defining Vector Subtraction
• Defining Scalar Multiplication of a Vector
• Standard Unit Vectors
• Writing Vectors in Component Form

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