# 1.4 Vectors  (Page 3/5)

 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 ).

$\mathbf{a}=a\stackrel{^}{a}$

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

$\stackrel{^}{a}=\frac{\mathbf{a}}{|\mathbf{a}|}$

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 $\stackrel{^}{i},\stackrel{^}{j}\mathrm{and}\stackrel{^}{k}$ . 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.

$\mathbf{a}=a\stackrel{^}{a}$

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

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

## 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

$\begin{array}{l}\mathbf{a}=-\mathbf{b}\\ ⇒\mathbf{a}+\mathbf{b}=0\\ \mathrm{and}\phantom{\rule{4pt}{0ex}}|\mathbf{a}|=|\mathbf{b}|\end{array}$

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.

pls explain what is dimension of 1in length and -1 in time ,what's is there difference between them
what are scalars
show that 1w= 10^7ergs^-1
what's lamin's theorems and it's mathematics representative
if the wavelength is double,what is the frequency of the wave
What are the system of units
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
58asagravitasnal firce
Amar
water boil at 100 and why
what is upper limit of speed
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
which colour has the shortest wavelength in the white light spectrum
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
x=5.8-3.22 x=2.58
what is the definition of resolution of forces
what is energy?
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin By Stephen Voron By Yasser Ibrahim By Tod McGrath By Marion Cabalfin By Nick Swain By OpenStax By OpenStax By Michael Sag By Yacoub Jayoghli By Jessica Collett