How to Perform Vector Addition and Subtraction

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

1. 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.
2. 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

1. Component-wise Addition: To add two vectors, combine their individual components.
   • \(v+w=(v_x​ + w_x​, v_y + w_y)\)
2. 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.
3. 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

1. Component-wise Subtraction: Subtracting vectors is analogous to addition but involves taking away magnitudes.
   • \(v+w=(v_x​ – w_x​, v_y – w_y)\)
2. 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 \(−w\).
3. 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, \(p−q=(3,−3)\).