Understanding Vector and Scalar Quantities in Physics

Representing Vectors

In mathematics and physics, vectors are often represented visually as directed line segments. In this representation, the length of the line segment corresponds to the magnitude of the vector, while the arrow indicates the direction.

In mathematical notation, vectors are typically written using lowercase letters that are either bold ($\mathbf{u}$) or underlined ($\underline{u}$).

Vector Components

While a directed line segment is useful for visualization, it is often more practical to describe a vector using its component form. This involves breaking the vector down into its constituent parts along the axes of a coordinate system.

A vector can be written in component form as: $\begin{pmatrix} x \\ y \end{pmatrix}$

In this notation:

• $x$ is the horizontal component.
• $y$ is the vertical component.

The direction of the vector is determined by the signs of these components:

• Positive $x$: Move to the right.
• Negative $x$: Move to the left.
• Positive $y$: Move up.
• Negative $y$: Move down.

Exam Tip: Always pay close attention to the signs of your components. A common mistake is to treat a negative component as a positive distance, which will lead to an incorrect direction in your final answer.

Calculating Magnitude

The magnitude of a vector, denoted as $|\mathbf{u}|$, represents the actual length of the directed line segment. Because the horizontal and vertical components form a right-angled triangle with the vector itself, we can calculate the magnitude using the Pythagoras Theorem.

For any vector with components $\begin{pmatrix} x \\ y \end{pmatrix}$, the formula for the magnitude is: $|\mathbf{u}| = \sqrt{x^2 + y^2}$

Worked Example: Finding Magnitude

Find the magnitude of the vector $\mathbf{v} = \begin{pmatrix} 12 \\ 5 \end{pmatrix}$.

Solution: Using the magnitude formula: $|\mathbf{v}| = \sqrt{12^2 + 5^2}$ $|\mathbf{v}| = \sqrt{144 + 25}$ $|\mathbf{v}| = \sqrt{169}$ $|\mathbf{v}| = 13$ The magnitude of vector $\mathbf{v}$ is 13.

Vector Operations

Addition of Vectors

To add two vectors geometrically, we use the tip-to-tail method. To add vector $\mathbf{v}$ to vector $\mathbf{u}$, you place the tail of vector $\mathbf{v}$ at the tip (the arrow end) of vector $\mathbf{u}$.

The resultant vector ($\mathbf{u} + \mathbf{v}$) is the single vector drawn from the starting tail of the first vector ($\mathbf{u}$) to the final tip of the second vector ($\mathbf{v}$).

When working with components, addition is very simple: you just add the corresponding horizontal and vertical parts: $\begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}$

Subtraction of Vectors

Subtraction can be thought of as adding the negative of a vector. The negative of a vector, $-\mathbf{u}$, has the exact same magnitude as $\mathbf{u}$ but points in the opposite direction.

Mathematically, subtracting $\mathbf{v}$ from $\mathbf{u}$ is written as: $\mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v})$

In component form, you simply subtract the components of the second vector from the first: $\begin{pmatrix} x_1 \\ y_1 \end{pmatrix} - \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix}$

Note: If a vector is multiplied by a negative scalar, its direction reverses. For example, if $\mathbf{u} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$, then $-\mathbf{u} = \begin{pmatrix} -3 \\ -2 \end{pmatrix}$.