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Solving problems is an essential part of the understanding process.

Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

Representative problems and their solutions

We discuss problems, which highlight certain aspects of the scalar vector product. For this reason, questions are categorized in terms of the characterizing features of the subject matter :

  • Angle between two vectors
  • Condition of perpendicular vectors
  • Component as scalar product
  • Nature of scalar product
  • Scalar product of a vector with itself
  • Evaluation of dot product

Angle between two vectors

Problem : Find the angle between vectors 2 i + j k and i k .

Solution : The cosine of the angle between two vectors is given in terms of dot product as :

cos θ = a . b ab

Now,

a . b = ( 2 i + j - k ) . ( 2 i - k )

Ignoring dot products of different unit vectors (they evaluate to zero), we have :

a . b = 2 i . i + ( - k ) . ( - k ) = 2 + 1 = 3 a = ( 2 2 + 1 2 + 1 2 ) = 6 b = ( 1 2 + 1 2 ) = 2 ab = 6 x 2 = ( 12 ) = 2 3

Putting in the expression of cosine, we have :

cos θ = a . b ab = 3 2 3 = 3 2 = cos 30 ° θ = 30 °

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Condition of perpendicular vectors

Problem : Sum and difference of two vectors a and b are perpendicular to each other. Find the relation between two vectors.

Solution : The sum a + b and difference a - b are perpendicular to each other. Hence, their dot product should evaluate to zero.

Sum and difference of two vectors

Sum and difference of two vectors are perpendicular to each other.

( a + b ) . ( a - b ) = 0

Using distributive property,

a . a - a . b + b . a - b . b = 0

Using commutative property, a.b = b.a , Hence,

a . a - b . b = 0 a 2 - b 2 = 0 a = b

It means that magnitudes of two vectors are equal. See figure below for enclosed angle between vectors, when vectors are equal :

Sum and difference of two vectors

Sum and difference of two vectors are perpendicular to each other, when vectors are equal.

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Component as scalar product

Problem : Find the components of vector 2 i + 3 j along the direction i + j .

Solution : The component of a vector “ a ” in a direction, represented by unit vector “ n ” is given by dot product :

a n = a . n

Thus, it is clear that we need to find the unit vector in the direction of i + j . Now, the unit vector in the direction of the vector is :

n = i + j | i + j |

Here,

| i + j | = ( 1 2 + 1 2 ) = 2

Hence,

n = 1 2 x ( i + j )

The component of vector 2 i + 3 j in the direction of “ n ” is :

a n = a . n = ( 2 i + 3 j ) . 1 2 x ( i + j )

a n = 1 2 x ( 2 i + 3 j ) . ( i + j ) a n = 1 2 x ( 2 x 1 + 3 x 1 ) a n = 5 2

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Nature of scalar product

Problem : Verify vector equality B = C , if A.B = A.C .

Solution : The given equality of dot products is :

A . B = A . C

The equality will result if B = C . We must, however, understand that dot product is not a simple algebraic product of two numbers (read magnitudes). The angle between two vectors plays a role in determining the magnitude of the dot product. Hence, it is entirely possible that vectors B and C are different yet their dot products with common vector A are equal.

We can attempt this question mathematically as well. Let θ 1 and θ 2 be the angles for first and second pairs of dot products. Then,

A . B = A . C

AB cos θ 1 = AC cos θ 2

If θ 1 = θ 2 , then B = C . However, if θ 1 θ 2 , then B C .

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Scalar product of a vector with itself

Problem : If | a + b | = | a b |, then find the angle between vectors a and b .

Solution : A question that involves modulus or magnitude of vector can be handled in specific manner to find information about the vector (s). The specific identity that is used in this circumstance is :

A . A = A 2

We use this identity first with the sum of the vectors ( a + b ),

( a + b ) . ( a + b ) = | a + b | 2

Using distributive property,

a . a + b . a + a . b + b . b = a 2 + b 2 + 2 a b cos θ = | a + b | 2 | a + b | 2 = a 2 + b 2 + 2 a b cos θ

Similarly, using the identity with difference of the vectors (a-b),

| a - b | 2 = a 2 + b 2 - 2 a b cos θ

It is, however, given that :

| a + b | = | a - b |

Squaring on either side of the equation,

| a + b | 2 = | a - b | 2

Putting the expressions,

a 2 + b 2 + 2 a b cos θ = a 2 + b 2 - 2 a b cos θ 4 a b cos θ = 0 cos θ = 0 θ = 90 °

Note : We can have a mental picture of the significance of this result. As given, the magnitude of sum of two vectors is equal to the magnitude of difference of two vectors. Now, we know that difference of vectors is similar to vector sum with one exception that one of the operand is rendered negative. Graphically, it means that one of the vectors is reversed.

Reversing one of the vectors changes the included angle between two vectors, but do not change the magnitudes of either vector. It is, therefore, only the included angle between the vectors that might change the magnitude of resultant. In order that magnitude of resultant does not change even after reversing direction of one of the vectors, it is required that the included angle between the vectors is not changed. This is only possible, when included angle between vectors is 90°. See figure.

Sum and difference of two vectors

Magnitudes of Sum and difference of two vectors are same when vectors at right angle to each other.

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Problem : If a and b are two non-collinear unit vectors and | a + b | = √3, then find the value of expression :

( a - b ) . ( 2 a + b )

Solution : The given expression is scalar product of two vector sums. Using distributive property we can expand the expression, which will comprise of scalar product of two vectors a and b .

( a - b ) . ( 2 a + b ) = 2 a . a + a . b - b . 2 a + ( - b ) . ( - b ) = 2 a 2 - a . b - b 2

( a - b ) . ( 2 a + b ) = 2 a 2 - b 2 - a b cos θ

We can evaluate this scalar product, if we know the angle between them as magnitudes of unit vectors are each 1. In order to find the angle between the vectors, we use the identity,

A . A = A 2

Now,

| a + b | 2 = ( a + b ) . ( a + b ) = a 2 + b 2 + 2 a b cos θ = 1 + 1 + 2 x 1 x 1 x cos θ

| a + b | 2 = 2 + 2 cos θ

It is given that :

| a + b | 2 = ( 3 ) 2 = 3

Putting this value,

2 cos θ = | a + b | 2 - 2 = 3 - 2 = 1

cos θ = 1 2 θ = 60 °

Using this value, we now proceed to find the value of given identity,

( a - b ) . ( 2 a + b ) = 2 a 2 - b 2 - a b cos θ = 2 x 1 2 - 1 2 - 1 x 1 x cos 60 °

( a - b ) . ( 2 a + b ) = 1 2

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Evaluation of dot product

Problem : In an experiment of light reflection, if a , b and c are the unit vectors in the direction of incident ray, reflected ray and normal to the reflecting surface, then prove that :

b = a - 2 ( a . c ) c

Solution : Let us consider vectors in a coordinate system in which “x” and “y” axes of the coordinate system are in the direction of reflecting surface and normal to the reflecting surface respectively as shown in the figure.

Reflection

Angle of incidence is equal to angle of reflection.

We express unit vectors with respect to the incident and reflected as :

a = sin θ i - cos θ j b = sin θ i + cos θ j

Subtracting first equation from the second equation, we have :

b - a = 2 cos θ j b = a + 2 cos θ j

Now, we evaluate dot product, involving unit vectors :

a . c = 1 x 1 x cos ( 180 ° - θ ) = - cos θ

Substituting for cosθ, we have :

b = a - 2 ( a . c ) c

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Questions & Answers

what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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Moment of inertia of a bar in terms of perpendicular axis theorem
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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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