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Scalar product (application)

Module by: Sunil Kumar Singh. E-mail the author

Summary: 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

Example 1

Problem : Find the angle between vectors 2i + jk and ik.

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

cos θ = a . b ab cos θ = a . b ab

Now,

a . b = ( 2 i + j - k ) . ( 2 i - k ) 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 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 ° cos θ = a . b ab = 3 2 3 = 3 2 = cos 30 ° θ = 30 °

Condition of perpendicular vectors

Example 2

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.

Figure 1: Sum and difference of two vectors are perpendicular to each other.
Sum and difference of two vectors
 Sum and difference of two vectors  (spq1.gif)

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

Using distributive property,

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

Figure 2: Sum and difference of two vectors are perpendicular to each other, when vectors are equal.
Sum and difference of two vectors
 Sum and difference of two vectors  (spq2.gif)

Component as scalar product

Example 3

Problem : Find the components of vector 2i + 3j 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 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 | n = i + j | i + j |

Here,

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

Hence,

n = 1 2 x ( i + j ) n = 1 2 x ( i + j )

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

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

Nature of scalar product

Example 4

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

Solution : The given equality of dot products is :

A . B = A . C 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 θ 1 and θ 2 θ 2 be the angles for first and second pairs of dot products. Then,

A . B = A . C A . B = A . C

AB cos θ 1 = AC cos θ 2 AB cos θ 1 = AC cos θ 2

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

Scalar product of a vector with itself

Example 5

Problem : If |a + b| = |ab|, 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 A . A = A 2

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

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

It is, however, given that :

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

Squaring on either side of the equation,

| a + b | 2 = | a - b | 2 | 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 ° 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.

Figure 3: Magnitudes of Sum and difference of two vectors are same when vectors at right angle to each other.
Sum and difference of two vectors
 Sum and difference of two vectors  (spq3.gif)

Example 6

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

It is given that :

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

Putting this value,

2 cos θ = | a + b | 2 - 2 = 3 - 2 = 1 2 cos θ = | a + b | 2 - 2 = 3 - 2 = 1

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

Evaluation of dot product

Example 7

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 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.

Figure 4: Angle of incidence is equal to angle of reflection.
Reflection
 Reflection  (spq4.gif)

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

Substituting for cosθ, we have :

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

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