Fundamentals of Mechanics/Units and Vectors/Vectors

Dimensional Analysis Vector Algebra When exploring physical quantities in nature, it is found that some quantities can be completely described by giving a number along with its unit, such as the mass of an object or the time between two events. These quantities are called scalar quantities. It is also found that other quantities are fully described by giving a number along with its unit in addition to a specified direction, such as the force on an object. These quantities are called vector quantities. Scalar quantities have magnitude but don't have a direction, and they obey the rules of ordinary arithmetic. Some examples with their typical variables are: mass: m volume: V temperature: T energy: E time: t Vector quantities have both magnitude and direction and obey the rules of vector algebra. Examples are displacement, force, velocity, and acceleration. Analytically, a vector can be indicated in a couple ways: Boldface: This notation is often used in printed material, but may be difficult for non-sighted folks to understand. Ex: x F v a Overarrow: This notation (used in this book) is the typical way to indicate a vector in handwriting and is more accessible in print and digital forms. Ex: x F v a The magnitude or length of a vector is indicated similar to how absolute value is indicated for scalars:  . Boldface:  Overarrow:  fig 1.1 frame|right|Fig. 1.1 A vector is represented geometrically by an arrow PQ Geometrically, a vector is represented by an arrow connecting two points (see Fig. 1.1). The length and direction of the arrow represent the magnitude and direction of the vector, respectively, and those values are independent of the choice of coordinate system. For vector A in Fig. 1.1, point P is called the initial point or tail of A, and Q is called the terminal point or head of A.