Chapter 2 Vectors
A physical world can be described by using varieties of physical quantities, they are a scalar physical quantity (or, simply scalars) and a vector physical quantity (or, simply vectors). A scalar quantity has only magnitude (size), e.g., mass, length, time, volume, density, work, speed, temperature, potential, current, resistance, flux, power, intensity, etc. and a vector quantity has both magnitude and direction, e.g., velocity, acceleration, displacement, momentum, force, weight, torque, impulse, moment, current density, etc. A quantity like surface area is a scalar in general, but when considered with its specific orientation, it is taken as a vector quantity. In physics, when area comes with any vector quantity in a mathematical operation it is considered as a vector. For instance, flux is a dot product of two vectors, one of them is an area. Area has a direction which is always taken as an outward normal to the surface. Mathematically, division of a vector with another vector is undefined, hence pressure is considered as a scalar even if it is defined as a force per unit area.
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Displacement: It is the shortest distance between any two points (e.g., distance between initial point and final point). Hence it has a particular direction. Velocity: If a direction is included with the speed then it is called a velocity. Weight: It is a force exerted by a planet on any object and hence is always pointed towards the center of the planet.
A vector is represented by an arrow. The arrow shows the direction of a vector and the length of the arrow represents the magnitude of that vector. It is denoted by a letter with an arrow over it such as \(\vec{OP} = \vec{a}\) and its magnitude is denoted by \(\vec{|a|}\) or simply by \(a\) as shown in Figure 2.0.1. A vector of unit magnitude is called a unit vector and is denoted by \(\hat{a}\text{,}\) a ’cap’, where \(\hat{a}=\frac{\vec{a}}{\vec{|a|}}\text{.}\) A vector which specifies the position of point P with respect to its origin O is called a position vector, where points O and P are known as the initial point (tail) and the terminal point (head) of the vector OP, respectively. The vectors with same magnitude but in opposite direction are termed as negative vectors. Vectors having the same direction are called like vectors, whereas those having opposite directions are called unlike vectors. The vectors parallel to the same line regardless of their magnitudes and directions are called collinear vectors. The vectors whose directions are not parallel to each other are non-collinear vectors. A system of vectors lying in the parallel planes or in the same plane are called coplanar vectors. The system of vectors which can not lie in the same plane or parallel planes are called non-coplanar vectors.
