5. VECTOR ALGEBRA

Welcome to Rajasthan Polytechnic. This blog is dedicated to providing detailed math notes to help you excel in your studies, whether you are preparing for polytechnic exams, competitive exams, or simply seeking to improve your understanding of mathematics.

The notes are organized chapter-wise to make learning more structured. Each chapter contains:

• Detailed Explanations of key concepts and formulas.
• Chapter-wise Video Links for visual learning and better understanding.
• PDF/Handwritten Notes that can be accessed at your convenience.

Additionally, you can join our WhatsApp and Telegram groups for easy access to PDF and Handwritten Notes as well as updates on new materials.

We hope this blog helps you in your academic journey.

📢 🔔 Important:
👉 Full PDF available in our WhatsApp Group | Telegram Channel
👉 Watch the full lecture on YouTube: BTER Polytechnic Classes

Vector Algebra Video Lecture - Watch Now

5. VECTOR ALGEBRA

---

5.1 Definition, Notation, and Rectangular Resolution of a Vector

What is a Vector?

A vector is a mathematical quantity that has both magnitude and direction. It is typically represented as an arrow, where the length of the arrow represents the magnitude, and the direction of the arrow indicates the direction of the vector.

Notation: Vectors are usually denoted by boldface letters, such as A, B, or $\vec{A}$ (when written in regular text).

Representation of a Vector:

In a two-dimensional space, a vector $\vec{A}$ can be represented by an ordered pair $(A_x, A_y)$, where:

• $A_x$ is the x-component of the vector (horizontal component).
• $A_y$ is the y-component of the vector (vertical component).

In three-dimensional space, a vector $\vec{A}$ is represented as $(A_x, A_y, A_z)$, where $A_z$ is the z-component (third dimension).

For example, a vector $\vec{A}$ in 2D may be written as:

$$\vec{A} = A_x \hat{i} + A_y \hat{j}$$

where $\hat{i}$ and $\hat{j}$ are the unit vectors along the $x$-axis and $y$-axis, respectively.

Rectangular Resolution of a Vector:

The process of resolving a vector into its components along the coordinate axes (typically the $x$, $y$, and $z$-axes in 3D space) is called rectangular resolution.

For a vector $\vec{A}$, if the magnitude is $A$ and the angle with the $x$-axis is $\theta$, the components of the vector in terms of the unit vectors $\hat{i}$ and $\hat{j}$ (in 2D) are:

$$A_x = A \cos \theta, \quad A_y = A \sin \theta$$

In 3D, the vector $\vec{A}$ can be resolved as:

$$\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$$

where $A_x$, $A_y$, and $A_z$ are the components of the vector along the $x$, $y$, and $z$-axes, respectively.

Example:

If a vector $\vec{A}$ has a magnitude of 5 units and makes an angle of $30^\circ$ with the $x$-axis in a 2D plane:

$$A_x = 5 \cos 30^\circ = 5 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}/2$$
$$A_y = 5 \sin 30^\circ = 5 \times \frac{1}{2} = 5/2$$

Thus, the vector $\vec{A}$ in component form is:

$$\vec{A} = \frac{5\sqrt{3}}{2} \hat{i} + \frac{5}{2} \hat{j}$$

---

5.2 Addition and Subtraction of Vectors

Vector Addition:

There are two main methods for adding vectors:

1. Graphical Method (Head-to-Tail Rule): To add two vectors $\vec{A}$ and $\vec{B}$, place the tail of vector $\vec{B}$ at the head of vector $\vec{A}$, and the resultant vector $\vec{R}$ is drawn from the tail of $\vec{A}$ to the head of $\vec{B}$.

2. Algebraic Method (Component Form): If two vectors $\vec{A} = A_x \hat{i} + A_y \hat{j}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j}$, their sum is given by:

$$\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$$

In 3D, if $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}$, the sum is:

$$\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j} + (A_z + B_z) \hat{k}$$

Example: Let $\vec{A} = 3\hat{i} + 4\hat{j}$ and $\vec{B} = 2\hat{i} - \hat{j}$. The sum is:

$$\vec{A} + \vec{B} = (3 + 2) \hat{i} + (4 - 1) \hat{j} = 5\hat{i} + 3\hat{j}$$

Vector Subtraction:

To subtract a vector $\vec{B}$ from $\vec{A}$, we can use the following rule:

$$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$$

This means that we add $\vec{A}$ and the negative of $\vec{B}$. The negative of a vector is obtained by reversing the direction of the vector (or multiplying it by $-1$).

If $\vec{A} = A_x \hat{i} + A_y \hat{j}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j}$, the difference is given by:

$$\vec{A} - \vec{B} = (A_x - B_x) \hat{i} + (A_y - B_y) \hat{j}$$

In 3D, if $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}$, the difference is:

$$\vec{A} - \vec{B} = (A_x - B_x) \hat{i} + (A_y - B_y) \hat{j} + (A_z - B_z) \hat{k}$$

Example: Let $\vec{A} = 3\hat{i} + 4\hat{j}$ and $\vec{B} = 2\hat{i} - \hat{j}$. The difference is:

$$\vec{A} - \vec{B} = (3 - 2) \hat{i} + (4 + 1) \hat{j} = \hat{i} + 5\hat{j}$$

---

Summary of Key Concepts and Formulas:

1. Definition of a Vector: A vector is represented by both magnitude and direction. Notation: A, B, or $\vec{A}$.

2. Rectangular Resolution:

   • In 2D, for a vector $\vec{A}$ making an angle $\theta$ with the $x$-axis: $$A_x = A \cos \theta, \quad A_y = A \sin \theta$$
     
   • In 3D, a vector $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$.

3. Vector Addition:

   • Graphical: Head-to-tail method.
   • Algebraic: $\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$.

4. Vector Subtraction:

   $$\vec{A} - \vec{B} = (A_x - B_x) \hat{i} + (A_y - B_y) \hat{j}$$

Thank You for Visiting Rajasthan Polytechnic! ✨

We trust that the notes provided on this blog will assist you in your studies. For those who prefer offline access to the materials, we offer PDF and Handwritten Notes through our WhatsApp and Telegram groups.

📢 🔔 Download PDF & Join Study Groups:
📥 WhatsApp Group: Join Now
📥 Telegram Channel: Join Now
📺 Watch Lecture on YouTube: BTER Polytechnic Classes
📍 Stay connected for more study materials! 🚀

Furthermore, we encourage you to explore the chapter-wise video tutorials for a deeper understanding of each topic.

To make full use of the resources, please refer to the following links:

• Join WhatsApp for PDF/Handwritten Notes:  WhatsApp link 
• Join Telegram for PDF/Handwritten Notes: Telegram link 
• Access Chapter-wise Video Tutorials:
  • Vector Algebra - Watch Now
  • Complete Playlist - Watch Now

Additional Resources:

• Official Rajasthan Polytechnic Website: Official Website link
• Study Resources and Notes: useful links

Thank you for visiting Rajasthan Polytechnic. We wish you the best of luck in your studies and look forward to assisting you in achieving your academic goals.