A Journey into Linear Algebra: Exploring the Basics of Vectors

6 min readDec 17, 2023

Linear algebra, a fundamental branch of mathematics, holds a pivotal position not only within its own field but also plays a crucial role in various disciplines such as physics, engineering, and computer science. It offers powerful tools for describing and solving problems ranging from simple geometric queries to complex algorithms in data analysis and machine learning.

The Role of Vectors in Mathematics and Physics

In the vast world of linear algebra, vectors are the most basic building blocks. Mathematically, vectors are objects that represent both magnitude and direction. In physics, they are used to depict forces, velocities, accelerations, and more. The concept of vectors extends beyond just two or three dimensions; in higher-dimensional spaces, vectors continue to be essential in representing direction and magnitude. Understanding the fundamental aspects of vectors is a crucial first step in mastering linear algebra.

In the following sections, we will delve into the fascinating world of vectors, exploring their definitions, representation methods, and basic operations. This foundational knowledge will pave the way for more advanced topics in linear algebra, such as matrices and systems of linear equations.

Defining and Representing Vectors

Concept of a Vector

In mathematics, a vector is defined as a geometric object that has both magnitude and direction. It can be visualized as a directed line segment, starting from one point (the origin) and ending at another (the terminal point). The “magnitude” of a vector typically refers to its length, while “direction” indicates the way it points.

Representation: Column and Row Vectors

Vectors can be represented in various ways, but the most common are as column vectors or row vectors. A column vector is composed of several values arranged in a vertical sequence, while a row vector arranges these values horizontally.

For example, a vector in two-dimensional space, v, can be represented as:
v = [x, y]
where x and y are the components of the vector in the horizontal (X-axis) and vertical (Y-axis) directions, respectively.

Vectors and Coordinate Systems

In different coordinate systems, the same vector can have different representations. In the Cartesian coordinate system, vectors are typically described by their components along the X and Y axes. In polar coordinates, vectors are described through their length and the angle with a reference direction.

Understanding how vectors are represented in different coordinate systems allows us to better apply them in mathematical and physical problems.

Basic Operations with Vectors

Vector Addition

Vector addition is a fundamental and important concept in linear algebra. When we add two vectors, we are essentially adding their corresponding components. For instance, if we have two vectors v = [x1, y1] and w = [x2, y2], their sum v + w would be [x1 + x2, y1 + y2].

Geometrically, this can be visualized as placing the start of one vector at the end of the other. The resulting vector’s starting point is the first vector’s origin, and its endpoint is the second vector’s endpoint.

Scalar Multiplication

Scalar multiplication, also known as scaling, is another fundamental operation. In scalar multiplication, we multiply a vector by a scalar (a regular number), which changes the vector’s magnitude but not its direction. For example, if there is a vector v = [x, y] and a scalar a, then a times v would be [ax, ay].

In physical terms, scalar multiplication can represent the increase or decrease in velocity or the scaling of a force.

Vector Subtraction and Negative Vectors

Vector subtraction is similar to vector addition but involves subtracting the components of vectors. If we have two vectors v = [x1, y1] and w = [x2, y2], then v — w would be [x1 — x2, y1 — y2].

A negative vector is the vector with the same magnitude as the original but in the opposite direction. For example, the negative of vector v would be represented as -v.

Vectors in Spatial Applications

Vectors as Representations of Direction and Displacement

In physics and engineering, vectors are widely used to represent direction and displacement. A vector not only tells us how far an object has moved but also in which direction it has moved. For instance, a vector [3, 4] indicates that an object has moved 3 units along the X-axis and 4 units along the Y-axis.

Application Examples in Physics

Vectors play a crucial role in physics. They are used to describe forces, velocities, accelerations, and more. For example, a force vector can inform us about the magnitude of the force and the direction in which it is applied, which is essential for solving physical problems and in engineering designs.

Vector Magnitude and Direction

Magnitude (Length) of a Vector

The magnitude or length of a vector is a crucial measure of its size. The length of a vector v = [x, y] in two-dimensional space can be calculated using the Pythagorean theorem, i.e., the square root of (x squared plus y squared). This length provides a measure of the vector irrespective of its direction.

Unit Vectors

A unit vector has a length of one. It is often used to represent direction since its length is normalized. Any non-zero vector can be converted into a unit vector by dividing it by its length, a process known as normalization.

Direction Angles of a Vector

The direction of a vector can be described by the angle it makes with a standard coordinate axis (usually the X-axis). For example, the direction of a vector in two-dimensional space can be determined by calculating the angle it makes with the X-axis, which can be found using inverse trigonometric functions.

Advanced Vector Concepts: Dot Product and Cross Product

Dot Product of Vectors

The dot product, also known as the scalar product, is a type of multiplication operation between two vectors, resulting in a scalar (a single number). For two vectors v = [x1, y1] and w = [x2, y2], their dot product is defined as x1x2 + y1y2. A key property of the dot product is its relation to the angle between the two vectors. The dot product is maximized when the vectors are in the same direction, minimized when they are in opposite directions, and is zero if they are perpendicular to each other.

The dot product has extensive applications in physics, such as in calculating the work done when a force is applied along a displacement.

Cross Product of Vectors

The cross product, also known as the vector product, is another operation that is specific to three-dimensional vectors. For two vectors v = [x1, y1, z1] and w = [x2, y2, z2], their cross product results in a new vector, which is perpendicular to the plane containing the original vectors. The length of this new vector is equal to the product of the lengths of the original vectors multiplied by the sine of the angle between them.

The cross product is particularly significant in physics, especially in problems involving torque and angular momentum.

Conclusion and Outlook

The Importance of Vectors in the Foundation of Linear Algebra

We have explored the fundamental concepts of vectors, from their definitions and representations to their applications in mathematics and physics. Understanding vectors is critical to grasping a wider range of linear algebra concepts, as vectors are the foundation for matrices, linear transformations, and multidimensional space analysis.

Preview of the Next Article: Matrices and Linear Equations

In our next article, we will delve deeper into another key topic in linear algebra: matrices and systems of linear equations. Matrices are not just a natural extension of vectors but also an indispensable tool in modern mathematics and physics. We will explore how to use matrices to solve systems of linear equations and discuss their applications in modeling and analyzing more complex systems.