Vectors are fundamental in mathematics and physics, representing quantities that have both magnitude and direction. In this section, we delve into the basics of vector representation, focusing on directed line segments. Vectors are not just abstract concepts but have practical applications in various fields, including engineering, physics, and computer science.

What is a Vector?

A vector is a mathematical entity characterized by magnitude and direction, represented graphically as an arrow. The arrow's length signifies the vector's magnitude, while its orientation shows the direction.

• Magnitude: The vector's length.
• Direction: The orientation of the vector.

Image courtesy of  Math Insight

Vectors in the Cartesian coordinate system can be denoted as $\mathbf{a} = (x, y)$, where $x$ and $y$ are its components along the horizontal and vertical axes, respectively.

Coordinate Representation of Vectors

In the Cartesian plane, a vector $\mathbf{a}$ is represented by $(x, y)$, marking its head's coordinates, with the tail typically at the origin $(0, 0)$.

• Tail: Starting point of the vector, often at $(0, 0)$.
• Head: Endpoint of the vector, located at $(x, y)$.

Image Courtesy of Nagwa

Worked Examples

Example 1: Calculating Magnitude

For a vector $\mathbf{a} = (3, 4)$:

• The magnitude $|\mathbf{a}|$ is calculated as $\sqrt{3^2 + 4^2} = 5$.

This demonstrates how to find the length of a vector using its components.

Example 2: Vector Addition

Given $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (1, 1)$, we find $\mathbf{c} = \mathbf{a} + \mathbf{b}$ by adding their components:

This process illustrates how to combine two vectors to obtain a resultant vector.

Example 3: Graphical Representation

To graphically represent $\mathbf{d} = (4, -2)$, plot an arrow starting at the origin and ending at $(4, -2)$. This graphical method helps visualize the vector's direction and magnitude.

Key Concepts

• Vectors have both magnitude and direction, represented as directed line segments.
• Coordinate representation allows easy manipulation and visualization of vectors.
• Adding vectors involves combining their respective components.