1.4 Vectors

A number of key fundamental physical concepts relate to quantities, which display directional property. Scalar algebra is not suited to deal with such quantities. The mathematical construct called vector is designed to represent quantities with directional property. A vector, as we shall see, encapsulates the idea of “direction” together with “magnitude”.

In order to elucidate directional aspect of a vector, let us consider a simple example of the motion of a person from point A to point B and from point B to point C, covering a distance of 4 and 3 meters respectively as shown in the Figure . Evidently, AC represents the linear distance between the initial and the final positions. This linear distance, however, is not equal to the sum of the linear distances of individual motion represented by segments AB and BC ( 4 + 3 = 7 m) i.e.

$$\begin{array}{c}\mathrm{AC}\ne \mathrm{AB}+\mathrm{BC}\end{array}$$

However, we need to express the end result of the movement appropriately as the sum of two individual movements. The inequality of the scalar equation as above is basically due to the fact that the motion represented by these two segments also possess directional attributes; the first segment is directed along the positive x – axis, where as the second segment of motion is directed along the positive y –axis. Combining their magnitudes is not sufficient as the two motions are perpendicular to each other. We require a mechanism to combine directions as well.

The solution of the problem lies in treating individual distance with a new term "displacement" – a vector quantity, which is equal to “linear distance plus direction”. Such a conceptualization of a directional quantity allows us to express the final displacement as the sum of two individual displacements in vector form :

$$\begin{array}{c}\mathbf{AC}=\mathbf{AB}+\mathbf{BC}\end{array}$$

The magnitude of displacement is obtained by applying Pythagoras theorem :

$$\begin{array}{c}\mathrm{AC}\surd \left({\mathrm{AB}}^2+{\mathrm{BC}}^2\right)=\surd \left(4^2+3^2\right)=\mathrm{5\; m}\end{array}$$

It is clear from the example above that vector construct is actually devised in a manner so that physical reality having directional property is appropriately described. This "fit to requirement" aspect of vector construct for physical phenomena having direction is core consideration in defining vectors and laying down rules for vector operation.

A classical example, illustrating the “fit to requirement” aspect of vector, is the product of two vectors. A product, in general, should evaluate in one manner to yield one value. However, there are natural quantities, which are product of two vectors, but evaluate to either scalar (example : work) or vector (example : torque) quantities. Thus, we need to define the product of vectors in two ways : one that yields scalar value and the other that yields vector value. For this reason product of two vectors is either defined as dot product to give a scalar value or defined as cross product to give vector value. This scheme enables us to appropriately handle the situations as the case may be.

$$\begin{array}{cc}W=\mathbf{F}\mathbf{.}\Delta \mathbf{r}& \text{……… Scalar dot product}\\ \mathbf{\tau }=\mathbf{r}x\mathbf{F}& \text{……… Vector cross product}\end{array}$$