1.4 Vectors  (Page 3/5)

Unit vector

Unit vector has a magnitude of one and is directed in a particular direction. It does not have dimension or unit like most other physical quantities. Thus, multiplying a scalar by unit vector converts the scalar quantity into a vector without changing its magnitude, but assigning it a direction ( Figure ).

$$\mathbf{a}=a\hat{a}$$

This is an important relation as it allows determination of unit vector in the direction of any vector " a as :

$$\hat{a}=\frac{\mathbf{a}}{\left|\mathbf{a}\right|}$$

Conventionally, unit vectors along the rectangular axes is represented with bold type face symbols like : $\mathbf{i},\mathbf{j}\mathrm{and}\mathbf{k},$ or with a cap heads like $\hat{i},\hat{j}\mathrm{and}\hat{k}$ . The unit vector along the axis denotes the direction of individual axis.

Using the concept of unit vector, we can denote a vector by multiplying the magnitude of the vector with unit vector in its direction.

$$\mathbf{a}=a\hat{a}$$

Following this technique, we can represent a vector along any axis in terms of scalar magnitude and axial unit vector like (for x-direction) :

$$\mathbf{a}=a\mathbf{i}$$

Other important vector terms

Null vector

Null vector is conceptualized for completing the development of vector algebra. We may encounter situations in which two equal but opposite vectors are added. What would be the result? Would it be a zero real number or a zero vector? It is expected that result of algebraic operation should be compatible with the requirement of vector. In order to meet this requirement, we define null vector, which has neither magnitude nor direction. In other words, we say that null vector is a vector whose all components in rectangular coordinate system are zero.

Strictly, we should denote null vector like other vectors using a bold faced letter or a letter with an overhead arrow. However, it may generally not be done. We take the exception to denote null vector by number “0” as this representation does not contradicts the defining requirement of null vector.

$$\mathbf{a}+\mathbf{b}=0$$

Negative vector

Negative vector

A negative vector of a given vector is defined as the vector having same magnitude, but applied in the opposite direction to that of the given vector.

It follows that if $\mathbf{b}$ is the negative of vector $\mathbf{a}$ , then

$$\begin{array}{l}\mathbf{a}=-\mathbf{b}\\ \Rightarrow \mathbf{a}+\mathbf{b}=0\\ \mathrm{and}\left|\mathbf{a}\right|=\left|\mathbf{b}\right|\end{array}$$

There is a subtle point to be made about negative scalar and vector quantities. A negative scalar quantity, sometimes, conveys the meaning of lesser value. For example, the temperature -5 K is a smaller temperature than any positive value. Also, a greater negative like – 100 K is less than the smaller negative like -50 K. However, a scalar like charge conveys different meaning. A negative charge of -10 μC is a bigger negative charge than – 5 μC. The interpretation of negative scalar is, thus, situational.

On the other hand, negative vector always indicates the sense of opposite direction. Also like charge, a greater negative vector is larger than smaller negative vector or a smaller positive vector. The magnitude of force -10 i N, for example is greater than 5 i N, but directed in the opposite direction to that of the unit vector i . In any case, negative vector does not convey the meaning of lesser or greater magnitude like the meaning of a scalar quantity in some cases.