Vector Concept

What Is a Vector?

To understand vectors, imagine a weather map showing wind direction and speed at various locations. On such maps, arrows are often drawn. These arrows indicate the direction of the wind and how strong it blows (usually shown by the length or thickness of the arrow).

In mathematics, quantities that have both magnitude (size) and direction like this are called vectors. Thus, the speed and direction of wind on a weather map is a good example of a vector. The length of the arrow represents the magnitude of the wind speed (for example, 30 km/h), and the direction of the arrow shows the direction in which it blows.

Drawing and Writing Vectors

Vectors are typically drawn as directed line segments (arrows).

• Vector 1: Direction of the vector from point $(0, 0)$ to point $(6, 8)$
• Vector 2: Direction of the vector from point $(5, 2)$ to point $(1, 1)$
• Vector 3: Direction of the vector from point $(4, 7)$ to point $(2, 0)$

Key Components of Vectors

1. Initial Point: The starting point of the vector.
2. Terminal Point: The ending point of the vector, marked with an arrowhead.
3. Magnitude: The length of the arrow, representing the value or size of the vector. Often also called vector length. The length of vector $\overrightarrow{AB}$ is denoted as $|\overrightarrow{AB}|$.
4. Direction: The direction pointed by the arrowhead.

Vector Notation

There are several ways to write vectors:

1. Two Capital Letters with an Arrow Above: Indicating the initial and terminal points. Example: $\overrightarrow{AB}$, meaning a vector from point A to point B.
2. One Lowercase Letter with an Arrow Above: Example: $\vec{a}$.
3. Bold Letters: Example: a or AB

Exercise

Are the shapes below vectors? Why?

These shapes are not vectors because:

1. Shape 1: because its line is curved, whereas vectors must be represented by straight lines.
2. Shape 2: because it consists of more than one straight line, whereas a vector must be represented by a single straight line.
3. Shape 3: because it has two directions (shown by arrows at both ends), whereas a vector must have only one clear direction.

A valid vector in mathematics must have these characteristics:

• It must be a straight line
• It must have one direction indicated by an arrow
• It must consist of a single straight line (not broken or curved)