Vector & Scalar Quantities

Graphical Representation: Vectors are visualized as directed line segments (arrows). The length of the arrow is proportional to the vector's magnitude, and the arrowhead points in the direction of the quantity.

Symbolic Notation: Vectors are typically written in boldface (e.g., A) or with an arrow above the letter (e.g., $\vec{A}$). Scalars are written in standard italics (e.g., $m$ for mass).

Component Form: A vector in a 2D plane can be broken down into two perpendicular parts: the x-component ($A_x$) and the y-component ($A_y$). This is expressed as $\vec{A} = A_x\hat{i} + A_y\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors.

The Resultant: The resultant is the single vector that represents the combined effect of two or more individual vectors. It is found by connecting the tail of the first vector to the tip of the last vector in a sequence.

Commutative Property: Vector addition is independent of order, meaning $\vec{A} + \vec{B} = \vec{B} + \vec{A}$. This is visually demonstrated by the parallelogram law of addition.

Vector Subtraction: Subtracting a vector is equivalent to adding its opposite. To find $\vec{A} - \vec{B}$, one adds a vector of the same magnitude as $\vec{B}$ but pointing in the exactly opposite direction ($180^\circ$ flip).

Resolution: This is the process of splitting a single vector into two or more components that act along specific axes. This simplifies complex problems by allowing independent analysis of horizontal and vertical motion.

Trigonometric Relations: For a vector $\vec{A}$ at an angle $\theta$ to the x-axis, the components are calculated as:

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

Reconstruction: To find the magnitude and direction from components, use the Pythagorean theorem and the tangent function:

• Magnitude: $A = \sqrt{A_x^2 + A_y^2}$
• Direction: $\theta = \tan^{-1}(\frac{A_y}{A_x})$

Distance vs. Displacement: Distance is a scalar representing the total path length, while displacement is a vector representing the straight-line change in position from start to finish.

Speed vs. Velocity: Speed is the scalar rate of motion, whereas velocity is the vector rate of motion in a specific direction.

Check the Units: Always ensure that you are adding quantities of the same type and unit. You cannot add a force vector to a velocity vector.

Directional Convention: Establish a clear coordinate system (e.g., Up/Right is positive) before starting calculations. Consistency in signs is the most common area for lost marks.

Sanity Check the Resultant: The magnitude of a resultant vector $\vec{R} = \vec{A} + \vec{B}$ must always fall between $|A - B|$ and $|A + B|$. If your calculated value is outside this range, re-check your trigonometry.

Component Method: For problems involving more than two vectors, always resolve them into x and y components rather than using graphical methods. Sum the components ($\\sum X, \\sum Y$) to find the final resultant.