AKT University Exam Special: Addition and Subtraction of Vectors Explained (Easy 10 Marks Guide)
📝 Table of Contents
Hinglish Version (AKTU B.Tech Exam ke liye)
Question: Vector ke Addition aur Subtraction ko Samjhaiye? (Explain Addition and Subtraction of Vectors?)
Dekho, B.Tech ke exam mein acche marks lane ke liye, Vector Addition aur Subtraction ka concept clear hona bahut zaroori hai. Isko hum do tarah se samajhte hain:
👉 1. Vector ka Jodna (Addition of Vectors)
Vector ko seedhe-seedhe (Like Scalars - $2+3=5$) nahi jod sakte. Iske liye do main rules hain:
A) Triangle Law of Vector Addition (head-to-tail) (त्रिभुज नियम)
- Kya hai? Agar do Vectors ($\vec{A}$ aur $\vec{B}$) ko ek triangle ki do sides ke roop mein dikhaya jaaye, jismein pehle ka head dusre ke tail se milta ho (order mein), toh unka **Resultant** ($\vec{R}$) us triangle ki teesri side (opposite order mein) dikhayegi.
- Formula: $\vec{R} = \vec{A} + \vec{B}$
- Magnitude (Resultant ki value) ka formula:
R = √(A² + B² + 2AB cosθ)Jahan $\theta$ (theta) $\vec{A}$ aur $\vec{B}$ ke beech ka angle hai.
B) Parallelogram Law of Vector Addition (समांतर चतुर्भुज नियम)
- Kya hai? Agar do Vectors ($\vec{A}$ aur $\vec{B}$) ko ek parallelogram ki do adjacent sides ke roop mein dikhaya jaaye, jo ki ek hi point se shuru ho rahe hon, toh unka **Resultant** ($\vec{R}$) us parallelogram ka main diagonal (jo ussi point se nikle) dikhayegi.
- Formula: Same as Triangle Law: $\vec{R} = \vec{A} + \vec{B}$.
- Magnitude: Same formula use hota hai.
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👉 2. Vector ka Ghatana (Subtraction of Vectors)
Vector Subtraction asal mein ek special type ka Addition hi hai.
- $\vec{A} - \vec{B}$ karne ka matlab hai: $\vec{A}$ mein **negative vector** $(-\vec{B})$ ko jodna.
- Formula: $\vec{R'} = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$
- Negative Vector ka Matlab: $-\vec{B}$ ka matlab hai woh Vector jiska magnitude (value) toh $\vec{B}$ ke barabar ho, par **direction** uske **bilkul opposite** ho (180° ulta).
- Magnitude (Resultant ki value) ka formula: Agar hum $\vec{A}$ aur $\vec{B}$ ke beech ke angle $\theta$ ko use karein, toh resultant magnitude yeh hoga:
R′ = √(A² + B² - 2AB cosθ)Notice karo: Addition mein $'+'$ tha, Subtraction mein $'-'$ hai.
English Version (AKTU B.Tech EE Exam Oriented)
Question: Explain addition and subtraction of vectors with relevant laws and formulas.
Vector operations, especially addition and subtraction, are fundamental concepts in Physics and Electrical Engineering. A thorough understanding is crucial for solving problems and scoring maximum marks in the B.Tech examination.
👉 1. Vector Addition
Unlike scalar quantities, vectors (which have both magnitude and direction) cannot be added arithmetically. Their addition follows specific geometric laws.
A) Triangle Law of Vector Addition (head-to-tail)
- Statement: If two vectors ($\vec{A}$ and $\vec{B}$) acting simultaneously at a point are represented in both magnitude and direction by the two sides of a triangle taken in order, then the **Resultant vector** ($\vec{R}$) is represented by the third side of the triangle taken in the **opposite order**.
- Mathematical Representation: $\vec{R} = \vec{A} + \vec{B}$
- Magnitude of the Resultant: If $\theta$ is the angle between $\vec{A}$ and $\vec{B}$, the magnitude $R$ is given by:
R = √(A² + B² + 2AB cosθ)
B) Parallelogram Law of Vector Addition
- Statement: If two vectors ($\vec{A}$ and $\vec{B}$) acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from that point, then the **Resultant vector** ($\vec{R}$) is represented in magnitude and direction by the **diagonal** of the parallelogram drawn from the same point.
- Mathematical Representation: $\vec{R} = \vec{A} + \vec{B}$
- Magnitude: The magnitude $R$ is calculated using the same formula as the Triangle Law:
R = √(A² + B² + 2AB cosθ).png)
👉 2. Vector Subtraction
Vector subtraction ($\vec{A} - \vec{B}$) is defined as the addition of vector $\vec{A}$ to the negative of vector $\vec{B}$.
- Concept: The negative of a vector $(-\vec{B})$ is a vector that has the **same magnitude** as $\vec{B}$ but acts in the **exactly opposite direction** (i.e., $180^\circ$ difference).
- Mathematical Representation: $\vec{R'} = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$
- Magnitude of the Resultant: When using the angle $\theta$ between the original vectors $\vec{A}$ and $\vec{B}$, the magnitude $R'$ of the difference is:
R′ = √(A² + B² - 2AB cosθ)📚 Download Notes & Study Material
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❓ Top 7 FAQs on Vector Operations
Here are answers to the most searched questions to help you revise quickly:
1. What is the fundamental difference between Vector Addition and Scalar Addition?
Answer: Scalar addition involves simple arithmetic addition (e.g., $2+3=5$). Vector addition, however, is geometric; it must account for the direction of the vectors, typically using the Triangle or Parallelogram law, resulting in the formula $\text{R} = \sqrt{(\text{A}^2 + \text{B}^2 + 2\text{AB} \cos\theta)}$.
2. When is the magnitude of the resultant vector maximum?
Answer: The magnitude is **maximum** when the two vectors are acting in the **same direction** ($\theta = 0^\circ$). In this case, $\cos\theta = 1$, and $R_{max} = A + B$.
3. When is the magnitude of the resultant vector minimum?
Answer: The magnitude is **minimum** when the two vectors are acting in **opposite directions** ($\theta = 180^\circ$). In this case, $\cos\theta = -1$, and $R_{min} = |A - B|$.
4. Can Vector Addition be commutative and associative?
Answer: **Yes,** Vector Addition is both **commutative** ($\vec{A} + \vec{B} = \vec{B} + \vec{A}$) and **associative** [$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$].
5. How does vector subtraction relate to addition?
Answer: Vector subtraction $\vec{A} - \vec{B}$ is mathematically the same as adding $\vec{A}$ to the negative of $\vec{B}$: $\vec{A} + (-\vec{B})$. The negative vector $(-\vec{B})$ simply has the same magnitude but the opposite direction.
6. What is the use of the formula $R' = \sqrt{(A^2 + B^2 - 2AB \cos\theta)}$?
Answer: This formula is used to find the magnitude ($R'$) of the **difference** between two vectors, $R' = |\vec{A} - \vec{B}|$, where $\theta$ is the angle between the **original** vectors $\vec{A}$ and $\vec{B}$.
7. How does this post help me score 10/10 in my AKTU B.Tech exam?
Answer: This post provides the **exact structure, clear definitions, mandatory laws (Triangle/Parallelogram), and crucial formulas** (Magnitude of Resultant and Difference) required by the university to secure full marks in a long-answer theoretical question on vector operations. The Hinglish explanation aids in quick memorization and conceptual clarity.
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