# 3: Vectors


"Philosophy is written in this grand book, the universe which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth." - Galileo Galilei, The Assayer, tr. Stillman Drake (1957), Discoveries and Opinions of Galileo pp. 237-8.

• 3.1: Vector Analysis
Certain physical quantities such as mass or the absolute temperature at some point in space only have magnitude. A single number can represent each of these quantities, with appropriate units, which are called scalar quantities. There are, however, other physical quantities that have both magnitude and direction. Force is an example of a quantity that has both direction and magnitude. 3 numbers are needed to represent the magnitude and direction of a vector quantity in a three dimensional space.
• 3.2: Coordinate Systems
Physics involve the study of phenomena that we observe in the world. In order to connect the phenomena to mathematics we begin by introducing the concept of a coordinate system. A coordinate system consists of four basic elements: (1) Choice of origin, (2) Choice of axes, (3) Choice of positive direction for each axis and (4) Choice of unit vectors at every point in space. There are three commonly used coordinate systems: Cartesian, cylindrical and spherical. In this chapter, we will describe a
• 3.3: Vectors
From the physicist’s point of view, we are interested in representing physical quantities such as displacement, velocity, acceleration, force, impulse, and momentum as vectors. We must always understand the physical context for the vector quantity. Thus, instead of approaching vectors as formal mathematical objects we shall instead consider the following essential properties that enable us to represent physical quantities as vectors.
• 3.4: Vector Product (Cross Product)

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