# How to Perform Vector Addition and Subtraction

Vector addition and subtraction are fundamental operations in vector algebra used to combine or differentiate vectors. Let's break down the concepts.

## Step-by-step Guide to Perform Vector Addition and Subtraction

Here is a step-by-step guide to perform vector addition and subtraction:

### Prelude: Grasping the Essence of Vectors

**Vector Definition**: A vector is a mathematical entity with both magnitude and direction. They are often depicted as arrows where the length represents the magnitude and the arrow’s direction denotes its direction.**Vector Components**: In a Cartesian coordinate system, a vector in two dimensions can be represented as \(v=(v_x, v_y)\), where \(v_x\) and \(v_y\) are the \(x\) and \(y\) components respectively.

### Chapter I: Vector Addition – The Confluence of Magnitudes and Directions

**Component-wise Addition**: To add two vectors, combine their individual components.- \(v+w=(v_x + w_x, v_y + w_y)\)

**Graphical Method**:**Initiation**: Draw the first vector starting from the origin.**Continuation**: From the head of the first vector, draw the second vector.**Conclusion**: The resultant vector (sum) starts from the origin and ends at the head of the second vector.

**Summarizing the Process**: The process is akin to walking a certain distance in one direction (first vector) and then continuing from there in another direction (second vector).

### Chapter II: Vector Subtraction – The Divergence of Pathways

**Component-wise Subtraction**: Subtracting vectors is analogous to addition but involves taking away magnitudes.- \(v+w=(v_x – w_x, v_y – w_y)\)

**Graphical Method**:**Initiation**: Begin by drawing the first vector, \(v\), starting from the origin.**Introduction of the Negative Vector**: Visualize or sketch the negative of the second vector, \(−w\). This is a vector of the same magnitude as \(w\) but in the opposite direction.**Continuation**: From the head of \(v\), draw \(−w\).**Conclusion**: The resultant vector (difference) begins at the origin and ends at the head of \(−w\).

**Summarizing the Process**: Imagine retracing a portion of a journey. You take your entire journey (first vector) and then move backward by the path denoted by the second vector.

### Postlude: Reflecting on Operations

- Vectors, with their dual nature of magnitude and direction, offer an elegant way to describe physical quantities. When we combine or separate vectors, we’re essentially juggling these two properties in a dance of mathematical harmony.
- Practicing these operations on varied vectors will solidify your understanding and enhance your ability to navigate the world of vector operations.

### Examples:

**Example 1:**

Given vectors \(a=(5,3)\) and \(b=(−2,7)\), find \(a+b\).

**Solution: **

\(a+b=(a_x + \ b_x, a_y + \ b_y)=(5+(−2),3+7) =(3,10)\)

So,** **\(a+b=(3,10)\).

**Example 2:**

Given vectors \(p=(7,2)\) and \(q=(4,5)\), find \(p−q\).

**Solution: **

To subtract vector \(q\) from vector \(p\), we subtract the respective components of the two vectors.\(p−q=(p_x−\ q_x, p_y−\ q_y)=(7−4,2−5)=(3,−3)\)

Thus, \(p−q=(3,−3)\).

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