EDEXCEL GCSE FOUNDATION MATHS - What are Vectors

Edexcel GCSE Foundation Maths: What are Vectors?

Introduction

Vectors are a powerful tool used to represent both direction and magnitude. They are different from scalars, which only have magnitude. Imagine a car driving north at 50 mph. This can be represented by a vector:

• Direction: North
• Magnitude: 50 mph

This tutorial will explore the basics of vectors, covering:

• Representing vectors
• Adding and subtracting vectors
• Scalar multiplication
• Applications of vectors

Representing Vectors

Vectors are often represented by arrows. The length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector.

Here's an example of a vector representing a force of 10 Newtons acting to the right:

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The arrow points to the right, indicating the direction of the force. The length of the arrow represents the magnitude of 10 Newtons.

Vectors can also be represented using column vectors:

[ ]

For example, the vector above can be written as:

[10 0]

The first number represents the horizontal component, and the second number represents the vertical component.

Adding and Subtracting Vectors

Vectors can be added and subtracted graphically or algebraically.

Graphical Addition:

1. Place the tail of the second vector at the head of the first vector.
2. Draw a new vector from the tail of the first vector to the head of the second vector. This new vector represents the sum of the two vectors.

Graphical Subtraction:

1. Reverse the direction of the second vector.
2. Add the reversed vector to the first vector using the graphical addition method.

Algebraic Addition/Subtraction:

1. Add/subtract the corresponding components of the vectors.

For example, to add the vectors:

[2 3] 
[1 4]

Simply add the corresponding components:

[2 + 1 3 + 4] = [3 7]

Scalar Multiplication

Multiplying a vector by a scalar (a number) changes the magnitude of the vector.

1. Multiply each component of the vector by the scalar.

For example, multiplying the vector:

[2 3]

by the scalar 2 gives:

[2 * 2 3 * 2] = [4 6]

The new vector has a magnitude twice as large as the original vector.

Applications of Vectors

Vectors are used in many different fields, including:

• Physics: Representing forces, velocities, and accelerations.
• Engineering: Designing structures, analyzing stresses and strains.
• Computer graphics: Creating animations and simulations.
• Navigation: Determining positions and directions.

Conclusion

This tutorial has provided an introduction to vectors, covering their representation, addition, subtraction, and scalar multiplication. Vectors are a fundamental concept in many branches of mathematics and science, and understanding them is essential for further exploration of these subjects.