4.2 Scalars and vectors

SCALARS AND VECTORS

In physics, we can classify quantities as scalars or vectors. A physical quantity with only magnitude is called a scalar. It is specified completely by a single number, along with the proper unit. Examples are the distance between two points (5 meters), the mass of an object( 2 kg), and the temperature (25 °C) of a body. Scalars can be added, subtracted, multiplied, and divided just like ordinary numbers. 

A vector quantity is a quantity that has both a magnitude and a direction and obeys the triangle law of addition or equivalently the parallelogram law of addition. So, a vector is specified by giving its magnitude by a number and its direction. Some physical quantities that are represented by vectors are displacement, velocity, acceleration, and force.

The vectors are denoted by putting an arrow over the symbols representing them. Thus, we write \(\overrightarrow{A B}\), \(\overrightarrow{B C}\) etc. Sometimes a vector is represented by a single letter such as \(\vec{v}, \vec{F}\) etc. Often in  books like NCERT, the vectors are represented by bold face letters like \(\mathbf{A B}, \mathbf{B C}, \mathbf{v}, \mathbf{f}\) etc.

EQUALITY OF VECTORS

Two vectors \(\mathbf{A}\) and \(\mathbf{B}\) are said to be equal if, and only if, they have the same magnitude and the same direction.

Figure 4.2(a) shows two equal vectors \(\mathbf{A}\) and B. We can easily check their equality. Shift B parallel to itself until its tail \(\mathrm{Q}\) coincides with that of \(A\), i.e. \(\mathrm{Q}\) coincides with O. Then, since their tips \(\mathrm{S}\) and \(\mathrm{P}\) also coincide, the two vectors are said to be equal. In general, equality is indicated as \(\mathbf{A}=\mathbf{B}\). Note that in Fig. 4.2(b), vectors \(\mathbf{A}^{\prime}\) and \(\mathbf{B}^{\prime}\) have the same magnitude but they are not equal because they have different directions. Even if we shift \(\mathbf{B}^{\prime}\) parallel to itself so that its tail \(Q^{\prime}\) coincides with the tail \(\mathrm{O}^{\prime}\) of \(\mathbf{A}^{\prime}\), the tip \(\mathrm{S}^{\prime}\) of \(\mathbf{B}^{\prime}\) does not coincide with the tip \(\mathrm{P}^{\prime}\) of \(\mathbf{A}^{\prime}\).