Dot Product and Vector

Introduction A qualitative or descriptive approach is generally not sufficient to obtain in-depth understanding of a physical phenomenon and the mechanisms leading to its manifestation. Instead, typically, the physicist will try to quantify the physical phenomenon by introducing certain physical quantities and by making use of mathematical tools. A physical quantity is an attribute of a physical object, system or phenomenon that can be quantified directly or indirectly from observations or measurements.

A standard of measurement is an accepted or approved instance or example of a quantity or quality against which others are judged or measured or compared. A unit is a particular physical quantity, defined and adopted by convention, against which other quantities of the same kind are compared to express their value. The measure of a physical quantity is obtained by comparing it to a unit. This process is called a measurement. Errors are intimately associated with any measurement. The value of a physical quantity is the product of its measure (numerical value) and the appropriate unit of measurement.

The dimension of a physical quantity gives a qualitative description of the latter in terms of products or quotients of the fundamental quantities (mass M, length L, time T, current intensity A, etc.. ) raised to appropriate powers. For example the dimension of charge per unit volume ? , denoted [? ], is ATL-3. The primary units are the kilogramme, the metre, the second and the ampere. Among physical quantities, two types can be distinguished: extensive quantities (e. g. volume, mass) whose value increases with amount and intensive quantities whose value is independent on amount (e. g.

density, concentration). Physical quantities can be represented by two main classes of mathematical objects, namely scalars and vectors. 2. Scalars Any physical quantity that has a magnitude (size or extent) but no direction is called a scalar quantity and is represented by a scalar. A scalar is a single positive or negative number. Scalars are subject to the laws of ordinary algebra. (commutativity, associativity, distributivity). Examples: mass, temperature, energy A scalar function (e. g. f (x, t)) evaluates to scalar value. In other words, its range is onedimensional. Dr S. Oree 3.

Vectors Mechanics I – 2013 Department of Physics University of Mauritius Any physical quantity that is fully characterized by a magnitude and a direction and which is subject to the parallelogram law of addition can be represented by a vector. It is called a vector quantity. Examples: Electric field, Force, displacement ? A vector is represented by a symbol topped by an arrow ( a ) or an underlined symbol ( a ). In books they may be represented by a bold symbol (a). A vector is represented on a diagram by a segment of straight line drawn to a chosen scale and terminated by an arrow.

The length of the segment represents magnitude (or absolute value) while the arrows represents direction. A vector has a line of action, a direction, a magnitude, a tail and a head. Direction, D Magnitude, M Tail, T Line of action, L Head, H Vector Figure 1: Vector representation ? ? ? The negative of a vector a is the vector ? a which has the same magnitude as a but is in the opposite direction. Vectors associated with a linear or directional effect are called polar vectors or simply vectors. Examples: Linear momentum, force, velocity. Vectors associated with axial rotation are called axial vectors or pseudo-vectors.

Examples: Torque, angular momentum, magnetic field vector at a point. Pseudo-vectors remain unchanged in direction under an inversion of the co-ordinate axes, while polar vectors reverse direction under such an operation. 3. 1. Unit vector A unit vector is a vector having a magnitude equal to unity. To distinguish it from other vectors, it is often represented by a symbol topped by a caret (or hat or chevron) symbol. ? i. e. u . ? ? Any vector A is equal to its magnitude A times the unit vector u parallel to it. ? ? A ? Au Dr S. Oree Mechanics I – 2013 Department of Physics

University of Mauritius 3. 2. Free and bound vectors A vector may refer to a physical quantity defined at a particular point in space that has meaning only at that point. Such a vector is referred to as a bound vector. (H, T, M, L, D are fixed). A vector that is not associated to a particular point in space is known as a free vector. (only M, and D are fixed). 3. 3. Collinear vectors Two vectors are said to be collinear if they have the same line of action or parallel lines of action. Note the two vectors are not necessarily in the same directions. They may also be in opposite directions.