Vectors and Scalars

Vectors and Scalars

• Scalars are quantities defined by magnitude only, described with a number and unit.
  • Examples include mass (30 kg), time (10 s), distance (10 m), and speed (50 mph).
• Vectors are quantities defined by both magnitude and direction.
  • Examples include velocity (30 m/s, North), force (10 N, downwards), and displacement (20 m, 30 degrees from the horizontal).

Vector Representation

• In textbooks, vectors are bold-faced letters (e.g., A or a).
• When writing, vectors have an arrow over the letter (e.g., $\overrightarrow{A}$ or $\overrightarrow{a}$).
• Graphically, vectors are arrows where length is proportional to magnitude and direction represents direction.

Properties of Vectors

• Equality of vectors requires the same magnitude and direction.
• The negative of a vector has the same magnitude but the opposite direction.

Vector Addition – Geometrical Methods

• Triangle Method: Place the tail of one vector at the head of the other; the resultant extends from the tail of the first to the head of the second.
• Parallelogram Method: Place tails of both vectors at the same point, construct a parallelogram, and the resultant extends from the common tail to the opposite vertex.

Vector Addition – Analytical Method

• Break each vector into components along orthogonal axes (x and y axes).
• Find the components of the resultant vector by adding corresponding components of individual vectors.

Example Problem: Car Displacement

Problem Statement

• A car travels 20.0 km due north and then 35.0 km in a direction $60.0^\circ$ west of north.
• The goal is to find the magnitude and direction of the resultant displacement.

Solution

Analytical Method Steps

• Choose coordinate system: $\hat{i}$ (east) and $\hat{j}$ (north)
• Vector A is (0 $\hat{i}$ + 20.0 $\hat{j}$) km
• Vector B is (-35.0 sin(60°) $\hat{i}$ + 35.0 cos(60°) $\hat{j}$) km which simplifies to (-30.3 $\hat{i}$ + 17.5 $\hat{j}$) km
• The resultant vector R = A + B which is (-30.3 $\hat{i}$ + 37.5 $\hat{j}$) km
• Magnitude of R is calculated as $\sqrt{(-30.3)^2 + (37.5)^2}$ km = 48.2 km
• Direction $\theta$ is calculated as $\tan^{-1} (\frac{30.3}{37.5}) = 38.9^\circ$