The physical quantities which we encounter in our daily lives may be broadly divided into two groups: scalars and vectors. A quantity which is completely specified by a number with an appropriate unit is called a scalar. It has only magnitude but no direction. Examples are mass, distance, time, speed, work, temperature, volume etc. A quantity which is completely specified by a number with an appropriate unit and a direction is called a vector. Thus, it has both magnitude and direction. Some examples of vectors are displacement, velocity, acceleration, force, momentum, torque, gravitational field etc.
Vector quantities follow the rules of vector algebra, which are different from the rules of arithmetic. This is an important criterion for checking whether a quantity is vector or not, as we shall see when we study Rotational Mechanics.
The graphical representation of a vector is a straight line with an arrowhead on one tip. The length of the line segment represents the magnitude of the vector on a convenient scale; the arrowhead gives the direction of the vector.
Three or more vectors which are in the same plane or are parallel to the same plane are called coplanar vectors. Equal vectors are those which have the same magnitude, same or parallel lines of support, and the same sense. The negative of a vector is the one whose magnitude is equal to that of the given vector, but the direction is opposite. A zero vector or null vector is the one which has zero magnitude and no specific direction. The position vector of a point represents the position of the point with respect to the origin of any coordinate system. The unit vector is obtained by dividing a vector by its magnitude. The magnitude of a unit vector is 1 (no unit), and its direction is the same as that of the original vector. As an example, if a force vector is 5 N directed southwards, the corresponding unit vector will be 1 due south.
The rules of addition of vectors are more complicated than those of arithmetic addition. The addition gives a resultant vector which produces the same effect as the combining vectors produce together. Two vectors can be added, both graphically and analytically, by the triangle law of vector addition or parallelogram law of vector addition. Several vectors can be added graphically by the polygon law of vector addition. For a more accurate addition of several vectors, analytically, we make use of addition of vectors by resolution method.
The subtraction of vectors follows similar approach as addition of vectors, with an important difference. The subtraction of one vector from another is defined as the addition of the negative of the vector to be subtracted. If P and Q are two vectors with the same unit then:
P - Q = P + (- Q)
The resolution of a vector is the process of determining a set of vectors which produce together the same effect as the original vector does alone. Each vector in the set is known as a component vector of the original vector. Thus, the process of resolution is opposite to the process of vector addition. Among many possibilities of resolution, the most useful ones are the resolution of a vector on a plane along two mutually perpendicular x- and y-axes AND the resolution of a vector in space along three mutually perpendicular x-, y- and z-axes.
Like addition and subtraction, the multiplication of vectors follows a set of rules different from those of arithmetic multiplication. The vectors to be multiplied need not have the same unit. There are two types of product known as the scalar product or dot product AND vector product or cross product.
The scalar product of two vectors is a scalar quantity equal to the product of the magnitudes of the given vectors and the cosine of the angle between them. The physical quantity "work" is the scalar product of force vector and displacement vector.
The vector product of two vectors is a vector quantity whose magnitude is equal to the product of their individual magnitudes and the sine of the smaller angle between them. The direction of the vector product is perpendicular to the plane containing the given vectors and its sense is determined by the right-hand corkscrew rule or the right-hand thumb rule, discussed in due course. The physical quantity "torque"is the vector product of displacement vector and force vector.