Graphical Vector Operations: Addition, Subtraction, and Scalar Multiplication

Problem Statement and Overview

• Problem Description: Given two vectors $\vec{a}$ and $\vec{b}$ shown in Figure 1, determine the graphical constructions and mathematical representations for:
  • Vector combination 1: $2\vec{a} + \vec{b}$
  • Vector combination 2: $\vec{b} - \vec{a}$

Characterization of Given Vectors

• Vector $\vec{a}$:
  • Orientation: Horizontal.
  • Direction: Directed toward the left.
  • Representation: An arrow extending from a vertical tail bar on the right to an arrowhead on the left.
• Vector $\vec{b}$:
  • Orientation: Tilted vertically / inclined.
  • Direction: Directed upward and slightly to the right.
  • Representation: An arrow extending from a crossbar tail at the bottom-left toward an arrowhead at the top-right.

Principles of Scalar Multiplication

• Definition: Scaling a vector $\vec{v}$ by a scalar factor $k$ produces a new vector $k\vec{v}$:
  • Magnitude: The magnitude of the scaled vector is ||k\vec{v}|| = |k| \times ||\vec{v}||\n - Direction for positive scalar (k > 0$): The direction of$k\vec{v}$is identical to that of$\vec{v}.\n - Direction for negative scalar (k < 0$): The direction of$k\vec{v}$is opposite to that of$\vec{v}$(rotated by$180^\circ).\n- Scaling Vector \vec{a}$by$k = 2:\n - The vector 2\vec{a}$maintains the exact same leftward horizontal direction as$\vec{a}.\n - The length (magnitude) of 2\vec{a}$is twice the original length of$\vec{a}.\n\n# Graphical Construction of 2\vec{a} + \vec{b}\n\n- Head-to-Tail (Tip-to-Tail) Method:\n - Step 1: Draw vector 2\vec{a}$horizontally pointing left, with a magnitude equal to double the length of$\vec{a}.\n - Step 2: Position the tail of vector \vec{b}$directly at the head (arrowhead tip) of vector$2\vec{a}.\n - Step 3: Draw vector \vec{b}$extending upward and slightly right from the tip of$2\vec{a}.\n - Step 4: Draw the resultant vector 2\vec{a} + \vec{b}$starting from the tail of$2\vec{a}$(the initial origin) directly to the head (tip) of vector$\vec{b}.\n- Parallelogram Method Alternative:\n - Place the tail of 2\vec{a}$and the tail of$\vec{b} at a common origin point.\n - Construct parallel sides matching 2\vec{a}$and$\vec{b} to form a complete parallelogram.\n - The resultant vector 2\vec{a} + \vec{b} extends as the diagonal from the common origin point to the opposite vertex of the parallelogram.\n\n# Graphical Construction of \vec{b} - \vec{a}\n\n- Algebraic Definition of Vector Subtraction:\n - Vector subtraction is expressed as the addition of the negative (opposite) vector:\n    \vec{b} - \vec{a} = \vec{b} + (-\vec{a})\n- Properties of Opposite Vector -\vec{a}:\n - Magnitude: Identical to the magnitude of \vec{a}.\n - Direction: Horizontally directed to the right (reversed relative to \vec{a}).\n- Head-to-Tail Construction Method:\n - Step 1: Draw vector \vec{b} in its original orientation (pointing upward and slightly right).\n - Step 2: Attach the tail of vector -\vec{a}$(pointing horizontally right) to the head of vector$\vec{b}.\n - Step 3: Draw the resultant vector \vec{b} - \vec{a}$starting from the tail of$\vec{b}$to the arrowhead tip of$-\vec{a}.\n- Tail-to-Tail Direct Difference Method:\n - Place the tail of \vec{a}$and the tail of$\vec{b} together at the exact same starting point.\n - The difference vector \vec{b} - \vec{a}$is the vector directed from the arrowhead tip of$\vec{a}$to the arrowhead tip of$\vec{b}$$.