Introduction to Vectors - Year 11 PDF Download (2024)

Introduction to Vectors (CIE IGCSE Maths: Extended)

Revision Note

Author

Expertise: Maths

Basic Vectors

What are vectors?

• A vector is a type of number that has both a size and a direction.
• We focus on two-dimensional vectors, although vectors can exist in any number of dimensions.

Representing vectors

• Vectors are represented as arrows, where the arrowhead shows the direction and the length of the arrow indicates the vector's magnitude (size).

Representing Vectors

• Print vectors are typically denoted by bold letters (e.g., vector 'a' as shown).
• Handwritten vectors are usually represented by underlined letters.

Alternative Representation

• Vectors can also be shown by indicating their starting and ending points with an arrow symbol on top.

Illustration:

• Order of letters in vectors is crucial; it determines direction.

Vectors in Transformation Geometry

In transformation geometry, translations are represented using column vectors.

Illustration:

• Multiplying a vector by a scalar

  • Definition: When you multiply a vector by a scalar, you are essentially scaling the vector by that scalar value.
  • Explanation: This operation involves multiplying each component of the vector by the scalar value.
  • Example: Let's consider a vector v = (2, 3). If we multiply this vector by a scalar 2, the result will be (4, 6). This means that each component of the vector is doubled.
  • Properties:
    • Multiplying a vector by 1 does not change the vector. The resulting vector is the same as the original vector.
    • Multiplying a vector by 0 results in a zero vector where both components are 0.
    • The direction of the vector remains the same when multiplied by a positive scalar but reverses when multiplied by a negative scalar.
    • The magnitude of the vector is scaled by the absolute value of the scalar.

Scalars and Vectors

• A scalar is a quantity that has only magnitude and no direction. In simpler terms, it is a regular number that you are accustomed to using.
• When a vector is multiplied by a positive scalar, the magnitude of the vector changes while its direction remains unchanged. This means that each component of the vector gets multiplied by the scalar.

Multiplying by Negative Scalars

• Multiplying a vector by a negative scalar not only changes the magnitude but also reverses the direction of the vector.

Vector Addition and Subtraction

• When adding two vectors, it is done geometrically by placing the tail of the second vector at the head of the first.

• Subtracting one vector from another is equivalent to adding the negative of the second vector to the first vector.

a - b = a + (-b)

Vectors Representation and Operations

• When vectors are displayed as column vectors, adding or subtracting involves manipulating the x and y coordinates.
• For instance, consider the vectors represented as column vectors where operations are based on adding or subtracting the x and y coordinates.

Illustrative Example

The points A, B, and C are plotted on a coordinate grid.

Vector Representation as Column Vectors

Begin by illustrating the three vectors on the grid:

• From A to B: Move 6 units to the right and 2 units upwards.
• From A to C: Move 7 units to the right and 6 units downwards.
• From C to B: Shift 1 unit to the left and 8 units upwards.

To confirm, utilize the column vectors obtained in the previous step.

Perform vector subtraction on the column vectors to validate the calculations.

Magnitude of a Vector

What is a vector?

• Vectors play crucial roles in mathematics. In mechanics, they symbolize velocity, acceleration, and forces. For IGCSE, vectors are integral in geometry, for instance, in translation. It's essential to grasp the Revision Notes on Vectors - Basics.
• Vectors possess both magnitude and direction. This discussion focuses on determining the magnitude or modulus of a vector, typically represented in column vector form.

Vectors

• In the realm of mechanics, vectors are representations of velocity, acceleration, and forces. In IGCSE, they are fundamental in geometric applications like translation. Understanding the Revision Notes on Vectors - Basics is imperative.

Vectors - Basics

• Magnitude and direction are inherent characteristics of vectors. This section delves into techniques for calculating the magnitude or modulus of a vector, commonly presented in column vector form.

Key Points

• Vectors are fundamental in mathematics and mechanics, representing essential physical quantities like velocity and forces.
• In IGCSE, vectors find applications in geometry, specifically in tasks such as translation.
• Understanding the basics of vectors, including their magnitude and direction, is crucial for various mathematical and scientific applications.

What is the magnitude or modulus of a vector?

Understanding Magnitude and Modulus

• When we talk about the magnitude or modulus of a vector, we are essentially referring to its size or length, which is always a positive value.
• For different types of vectors, such as velocity or force, the magnitude represents specific properties:
  • For velocity, the magnitude corresponds to speed.
  • For a force, the magnitude indicates the strength of the force in Newtons.
• In the context of vectors, the terms "magnitude" and "modulus" are interchangeable and signify the same concept.
• From a geometric perspective, the magnitude or modulus of a vector represents its distance, always being a non-negative value regardless of direction.
• The direction of the vector does not influence its magnitude or modulus; only the size matters.
• Mathematically, the magnitude or modulus of a vector is denoted by vertical lines, such as | a |, indicating the magnitude of vector 'a'.

This HTML output presents a detailed explanation of the concept of vector magnitude and modulus, illustrating their significance and applications in various contexts.

Magnitude or Modulus of a Vector

The magnitude or modulus of a vector is indicated by vertical lines. For example, | a | would represent the magnitude of vector a.

• Magnitude is denoted by vertical lines: | a | represents the magnitude of vector a.

Finding the Magnitude or Modulus of a Vector

To find the magnitude or modulus of a vector, you can use Pythagoras' Theorem.

• Pythagoras' Theorem can be applied to find the magnitude of a vector.

One way to find the magnitude of a vector is by sketching it to form a right-angled triangle, even if not to scale.

An Example for Finding Vector Magnitude

Let's consider an example of finding the magnitude of a vector using the Pythagorean theorem.

Test your understanding by trying similar problems. Move on to the next topic when you're ready for more!

Work hard and aim for better grades. You can download notes on Introduction to Vectors for further learning.

Explore Courses

Continue with Google

Track your progress, build streaks, highlight & save important lessons and more!