Further Pure Mathematics | Scalar and vector quantities

Welcome to Scalar and Vector Quantities!

Hello future Further Pure Mathematicians! This chapter introduces one of the most fundamental concepts in advanced mathematics and physics: Vectors. Don't worry if this seems tricky at first—we are simply learning how to describe movement and forces much more accurately than before.

In this chapter, we will learn to distinguish between quantities that only have size (scalars) and those that have both size and direction (vectors). Understanding this difference is absolutely crucial for success in mechanics and higher-level mathematics.

1. Understanding Magnitude and Direction

Before diving into vectors, we need to define the two core components used to describe physical quantities:

Magnitude

Magnitude simply means the size, amount, or value of a quantity. It is represented by a single number, often with units (like 5 kilograms, 10 seconds, or 50 miles per hour).

Direction

Direction tells you which way a quantity is acting (e.g., North, East, Up, or at an angle of \(30^\circ\) from the horizontal).

2. Scalars vs. Vectors: The Essential Difference

All quantities in mathematics and physics can be classified into two groups based on whether or not they require direction for their full description.

A) Scalar Quantities

A Scalar Quantity is defined by its magnitude only. Direction is either irrelevant or non-existent for a scalar quantity.

• Key Feature: Size only.
• Example Analogy: If you say you need 5 kilograms of flour, the direction doesn't matter.

Examples of Scalars:

• Time (2 hours)
• Mass (50 kg)
• Distance (10 km travelled)
• Speed (120 km/h)
• Temperature (25 °C)

B) Vector Quantities

A Vector Quantity is defined by both magnitude and direction.

• Key Feature: Size AND Direction.
• Example Analogy: If a GPS tells you to travel 10 km North-East, the direction is vital.

Examples of Vectors:

• Displacement (change in position)
• Velocity (speed in a specific direction)
• Force (a push or pull in a specific direction)
• Acceleration

Key Takeaway: If direction matters for the quantity to make sense, it's a vector. If only the amount matters, it's a scalar.

3. Representing Vectors

Vectors can be shown visually or written algebraically. We need clear notation to avoid confusing them with scalars.

A) Geometric Representation (Directed Line Segment)

Visually, a vector is represented by an arrow (a directed line segment).

• The length of the arrow represents the magnitude of the vector.
• The arrowhead indicates the direction.
• If a vector goes from point A to point B, we write it as \(\vec{AB}\).

B) Algebraic Representation (Column Vectors)

In coordinates, we often use a column vector to represent the movement in the x and y directions.

A vector \(\mathbf{a}\) is usually written as:

$$ \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} $$

• \(x\) represents the horizontal movement (positive is right, negative is left).
• \(y\) represents the vertical movement (positive is up, negative is down).

Example: The vector \(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\) means move 3 units right and 1 unit down.

C) Notation

Since direction is important, we must distinguish vectors from simple numbers (scalars) when writing them down:

• In print (like textbooks), vectors are usually written in bold lowercase letters (e.g., \(\mathbf{a}, \mathbf{b}\)).
• When writing by hand, we usually place a line or arrow above the letter (e.g., \(\vec{a}\) or \(\underline{a}\)).

4. Vector Arithmetic: The Algebra of Movement

Unlike scalars, adding and subtracting vectors is not always straightforward addition or subtraction of their magnitudes. We combine their components.

A) Addition of Vectors

1. Algebraically (using Column Vectors)

To add two vectors, you simply add their corresponding components.

If \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\), then:

$$ \mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix} $$

Example: If \(\mathbf{a} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\), then \(\mathbf{a} + \mathbf{b} = \begin{pmatrix} 5 + (-1) \\ 2 + 4 \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}\).

2. Geometrically (The Triangle Law)

To add \(\mathbf{a} + \mathbf{b}\) visually, we use the Triangle Law:

1. Draw vector \(\mathbf{a}\).
2. Start vector \(\mathbf{b}\) where vector \(\mathbf{a}\) ends (head-to-tail).
3. The resultant vector, \(\mathbf{a} + \mathbf{b}\), is the vector drawn from the start of \(\mathbf{a}\) to the end of \(\mathbf{b}\).

Did you know? This resultant vector is often called the net displacement or resultant force.

B) Subtraction of Vectors

Subtracting a vector is the same as adding its negative.

The vector \(-\mathbf{a}\) has the same magnitude as \(\mathbf{a}\) but acts in the opposite direction.

$$ \mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b}) $$

To subtract algebraically, subtract the corresponding components:

$$ \mathbf{a} - \mathbf{b} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix} $$

Example: If \(\mathbf{a} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}\), then \(\mathbf{a} - \mathbf{b} = \begin{pmatrix} 5 - 1 \\ 2 - 4 \end{pmatrix} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}\).

C) Multiplication by a Scalar

Multiplying a vector by a scalar \(k\) changes the magnitude, and potentially the direction, but the components remain in the same ratio.

If \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\), then:

$$ k\mathbf{a} = \begin{pmatrix} kx \\ ky \end{pmatrix} $$

What does \(k\) do?

• If \(|k| > 1\): The vector is stretched (magnitude increases).
• If \(0 < |k| < 1\): The vector is shrunk (magnitude decreases).
• If \(k\) is positive: The direction remains the same.
• If \(k\) is negative: The direction is reversed (it points exactly the opposite way).

Example: If \(\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\), then \(4\mathbf{a} = \begin{pmatrix} 12 \\ 4 \end{pmatrix}\) (four times longer in the same direction). And \(-2\mathbf{a} = \begin{pmatrix} -6 \\ -2 \end{pmatrix}\) (twice as long, reversed direction).

Key Takeaway: Vector addition follows the "head-to-tail" rule geometrically, but algebraically, we just combine the components.

5. Finding the Magnitude of a Vector

Since a vector in 2D space is essentially a diagonal line on a coordinate plane, we can find its length (its magnitude) using Pythagoras' theorem.

Notation for Magnitude

The magnitude of vector \(\mathbf{a}\) is denoted by \(|\mathbf{a}|\) or sometimes \(||\mathbf{a}||\). Think of the lines as absolute value signs—we are only interested in the size (length), which must be positive.

The Magnitude Formula

If the vector is \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\), the magnitude is:

$$ |\mathbf{a}| = \sqrt{x^2 + y^2} $$

Step-by-Step Example:

Find the magnitude of vector \(\mathbf{v} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\).

1. Identify the components: \(x = 4\) and \(y = -3\).
2. Square the components: \(x^2 = 4^2 = 16\); \(y^2 = (-3)^2 = 9\).
3. Add the squared components: \(16 + 9 = 25\).
4. Take the square root: \(|\mathbf{v}| = \sqrt{25} = 5\).

The magnitude of \(\mathbf{v}\) is 5 units.

6. Parallel Vectors and Collinearity

Two vectors are parallel if they act in the same direction or in exactly opposite directions. We can determine if they are parallel by checking if one is a scalar multiple of the other.

The Test for Parallelism

Vector \(\mathbf{a}\) is parallel to vector \(\mathbf{b}\) if and only if:

$$ \mathbf{a} = k\mathbf{b} $$

where \(k\) is a scalar constant (any real number except zero).

• If \(k > 0\), they point in the same direction.
• If \(k < 0\), they point in opposite directions.

Example: If \(\mathbf{p} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\). Since \(\mathbf{p} = 3\mathbf{q}\), the vectors are parallel.

Collinearity

A crucial application of parallel vectors is proving that three points (A, B, C) are collinear (meaning they lie on the same straight line).

To prove points A, B, and C are collinear, you must show two things:

1. The vector \(\vec{AB}\) is parallel to the vector \(\vec{BC}\). (i.e., \(\vec{AB} = k \vec{BC}\))
2. The vectors share a common point (in this case, point B).

Because they are parallel and share a point, they must lie on the same line.

Key Takeaway: Parallel vectors are simply scaled versions of one another.