Group: 661a1 Lesson: Technical Foreign Language

Student: Rəsul Cəfərzadə
Group: 661a1
Lesson: Technical Foreign Language
Teacher: Kerimova Shuse
Topic: Vectors
Vectors are mathematical quantities that have both magnitude and direction. They are used to represent a wide range of physical quantities, such as displacement, velocity, force, and more. Vectors are typically represented graphically as arrows, where the length of the arrow represents the magnitude, and the direction of the arrow represents the direction of the vector.
What are Vectors?
In contrast to scalars, which have only magnitude (e.g., mass, temperature), vectors provide a more complete description of physical phenomena by including information about both how much and in which direction something is changing or moving. Vectors play a fundamental role in various fields, including physics, engineering, computer science, and many other areas where understanding and manipulating quantities with direction is important.
Components of a Vector
The components of a vector are the individual parts into which a vector can be broken down, typically in a coordinate system. The number of components depends on the dimension of the vector (2D or 3D, for example).

• 2D Vectors:

- In a two-dimensional vector, you have two components: one for the horizontal (x-axis) and one for the vertical (y-axis) direction.
- A 2D vector is often represented as \((x, y)\), where \(x\) is the horizontal component, and \(y\) is the vertical component.

2. 3D Vectors:
- In a three-dimensional vector, you have three components: one for the x-axis, one for the y-axis, and one for the z-axis.
- A 3D vector is represented as \((x, y, z)\), where \(x\), \(y\), and \(z\) represent the components along the respective axes.
These components are crucial for performing vector operations, such as vector addition, subtraction, and scalar multiplication. They allow you to analyze and manipulate vectors in a coordinate system, making them essential in various scientific and engineering applications.
Vector operations
Vector operations are fundamental mathematical manipulations used to work with vectors in a variety of fields, including physics, engineering, computer science, and more. The most common vector operations include:
1. Vector Addition: When two or more vectors are added together, their effects are combined. You can add vectors graphically by connecting them tip-to-tail, or algebraically by adding their components. For example, if you have two 2D vectors \((x1, y1)\) and \((x2, y2)\), you can add them as \((x1 + x2, y1 + y2)\).
2. Vector Subtraction: Vector subtraction is the reverse of addition. To subtract vector B from vector A, you add the negative of vector B to vector A. Algebraically, you can subtract vectors by subtracting their components.
3. Scalar Multiplication: You can scale a vector by multiplying it by a scalar (a single number). This operation changes the magnitude of the vector without affecting its direction. For example, if you multiply a vector \((x, y)\) by a scalar \(c\), you get \((cx, cy)\).
4. Dot Product (Scalar Product): The dot product of two vectors A and B is a scalar value that measures the cosine of the angle between them. Algebraically, the dot product is calculated as \(A · B = |A||B| cos(θ)\), where \(|A|\) and \(|B|\) are the magnitudes of A and B, and \(θ\) is the angle between them.
5. Cross Product (Vector Product): The cross product of two 3D vectors A and B results in a new vector that is perpendicular to both A and B. The magnitude of the cross product is equal to the product of the magnitudes of A and B and the sine of the angle between them. The direction of the resulting vector follows the right-hand rule.
Unit vectors i, j, and k are commonly used in three-dimensional Cartesian coordinate systems to represent the direction of vectors along the x, y, and z-axes, respectively. These unit vectors have a magnitude of 1 and point in the positive direction of their respective axes. Here's a brief explanation of each:
Vector Notation

• Unit Vector i:
• - i represents the unit vector along the x-axis.
• - It points in the positive direction of the x-axis.

4. - Mathematically, i = (1, 0, 0) in 3D Cartesian coordinates.

2. Unit Vector j:
- j represents the unit vector along the y-axis.
- It points in the positive direction of the y-axis.
- Mathematically, j = (0, 1, 0) in 3D Cartesian coordinates.
3. Unit Vector k:
- k represents the unit vector along the z-axis.
- It points in the positive direction of the z-axis.
- Mathematically, k = (0, 0, 1) in 3D Cartesian coordinates.
Unit vectors are helpful for describing the direction of vectors in 3D space and for expressing vectors in terms of their components. For example, you can represent a vector v as a sum of its components using these unit vectors, such as v = (2i + 3j - k), which means the vector has a component of 2 in the x-direction, 3 in the y-direction, and -1 in the z-direction.
Thanks for attentions