EDEXCEL GCSE HIGHER MATHS - What are Vectors and Their Operations

Edexcel GCSE Higher Maths: Vectors and Their Operations

Introduction

Vectors are quantities that have both magnitude (size) and direction. They are often represented by arrows, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction.

Representing Vectors

Vectors can be represented in a few different ways:

• Geometrically: Using an arrow with a specific length and direction.
• Algebraically: Using a lowercase letter with an arrow above it, for example, $\overrightarrow{a}$.
• Component form: Using a column vector, for example, $\begin{pmatrix} 2 \ 3 \end{pmatrix}$. This represents a vector that moves 2 units to the right and 3 units upwards.

Operations on Vectors

Addition

To add vectors, we can use the parallelogram law:

1. Place the vectors head to tail.
2. Draw the diagonal of the parallelogram formed by the vectors. This diagonal represents the sum of the vectors.

Alternatively, we can add vectors by adding their corresponding components:

    a + b =  (a1 + b1) 
             (a2 + b2)

Subtraction

To subtract vectors, we can think of subtracting a vector as adding the negative of that vector. The negative of a vector has the same magnitude but the opposite direction.

Alternatively, we can subtract vectors by subtracting their corresponding components:

    a - b = (a1 - b1)
             (a2 - b2)

Multiplication by a Scalar

Multiplying a vector by a scalar (a number) changes its magnitude but not its direction:

• If the scalar is positive, the magnitude is increased.
• If the scalar is negative, the magnitude is increased and the direction is reversed.

Examples

1. Adding Vectors

Let $\overrightarrow{a} = \begin{pmatrix} 2 \ 1 \end{pmatrix}$ and $\overrightarrow{b} = \begin{pmatrix} -1 \ 3 \end{pmatrix}$. Find $\overrightarrow{a} + \overrightarrow{b}$.

    a + b =  (2 + (-1))
             (1 + 3) 
          =  (1)
             (4)

2. Subtracting Vectors

Let $\overrightarrow{c} = \begin{pmatrix} 4 \ 2 \end{pmatrix}$ and $\overrightarrow{d} = \begin{pmatrix} 1 \ -1 \end{pmatrix}$. Find $\overrightarrow{c} - \overrightarrow{d}$.

    c - d =  (4 - 1)
             (2 - (-1))
          =  (3)
             (3)

3. Multiplying by a Scalar

Let $\overrightarrow{e} = \begin{pmatrix} 3 \ 2 \end{pmatrix}$. Find $2\overrightarrow{e}$.

    2e =  2 * (3)
          2 * (2)
        =  (6)
           (4)

Applications of Vectors

Vectors are used in many areas of mathematics, physics, and engineering, including:

• Navigation: Representing displacement, velocity, and acceleration.
• Forces: Representing forces acting on an object.
• Geometry: Representing lines, planes, and other geometric objects.

Conclusion

Understanding vectors and their operations is crucial for success in higher-level mathematics. This tutorial provides a basic introduction to the concept of vectors and their operations. Further exploration of the topic will cover more complex operations and applications.