Vector addition and subtraction Knowledge-Based Study

Study notes

1. In physics, quantities are categorized based on the information needed to describe them completely:

   • Scalar quantities: Specified completely by a numerical value (magnitude) and an appropriate unit, such as mass (5 kg), time (12 s), or temperature (20 °C).
   • Vector quantities: Require both a magnitude and a direction in space, such as displacement, velocity, and force.

   A vector is represented geometrically by a directed line segment (an arrow). The length of the arrow is proportional to the magnitude of the vector, while the arrowhead points in the direction of the vector. Symbolically, a vector is written with an arrow above the letter, such as $\vec{A}$, or in boldface, such as A, and its magnitude is denoted as $|\vec{A}|$ or simply $A$.

   The sum of two or more vectors is called the resultant vector, denoted as $\vec{R}$. Unlike scalars, vectors cannot be added using ordinary arithmetic unless they are parallel and share the same line of action.

   1. The Head-to-Tail Method

   To find $\vec{R} = \vec{A} + \vec{B}$:

   1. Draw vector $\vec{A}$ to scale in the correct direction.
   2. Place the tail (initial point) of vector $\vec{B}$ at the head (terminal point) of vector $\vec{A}$.
   3. Draw the resultant vector $\vec{R}$ from the tail of $\vec{A}$ directly to the head of $\vec{B}$.

   2. The Parallelogram Method

   If two vectors $\vec{A}$ and $\vec{B}$ start from the same common point (tail-to-tail), construct a parallelogram using $\vec{A}$ and $\vec{B}$ as adjacent sides. The diagonal extending from the common tail to the opposite vertex represents the resultant vector $\vec{R} = \vec{A} + \vec{B}$.

   Vector addition is commutative ($\vec{A} + \vec{B} = \vec{B} + \vec{A}$) and associative ($(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$).

   Vector subtraction is defined as the addition of an opposite (negative) vector. The negative of a vector $\vec{B}$, written as $-\vec{B}$, is defined as a vector having the exact same magnitude as $\vec{B}$, but pointing in the opposite direction (rotated by 180°).

   Thus, subtracting vector $\vec{B}$ from vector $\vec{A}$ is expressed as:

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

   Steps for Graphical Subtraction:

   1. Identify vector $\vec{B}$ and reverse its direction completely to obtain $-\vec{B}$.
   2. Add vector $-\vec{B}$ to vector $\vec{A}$ using the standard head-to-tail method.
   3. Draw the resultant difference vector from the tail of $\vec{A}$ to the head of $-\vec{B}$.

   Note that vector subtraction is not commutative: $\vec{A} - \vec{B} \neq \vec{B} - \vec{A}$. In fact, $\vec{A} - \vec{B} = -(\vec{B} - \vec{A})$.

   While graphical methods provide geometric intuition, analytical methods using rectangular components provide exact numerical results.

   In a two-dimensional Cartesian coordinate system, any vector $\vec{A}$ making an angle $\theta$ with the positive x-axis can be resolved into two perpendicular components:

   • $A_x = A \cos\theta$
   • $A_y = A \sin\theta$

   Adding and Subtracting by Components

   When vectors are added or subtracted, their respective components combine independently:

   For addition, $\vec{R} = \vec{A} + \vec{B}$:

   • $R_x = A_x + B_x$
   • $R_y = A_y + B_y$

   For subtraction, $\vec{D} = \vec{A} - \vec{B}$:

   • $D_x = A_x - B_x$
   • $D_y = A_y - B_y$

   Once the resultant components are calculated, the magnitude and direction of the resultant vector $\vec{R}$ are found using the Pythagorean theorem and trigonometry:

   $$R = \sqrt{R_x^2 + R_y^2}$$

   $$\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)$$