Dot & cross product of vectors

Dot & Cross product of vectors

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What is dot product?

o The dot product of two vectors A and

B is defined as the scalar value AB cosθ, where θ is the angle between them such that 0≤θ≤π.

What is dot product?

o It is denoted by A.B by placing a dot

sign between the vectors. o So we have the equation,

A.B = AB cosθo Another name of dot product is

scalar product.

What is cross product?

o The cross product of two vectors A

and B is defined as AB sinθ with a direction perpendicular to A and B in right hand system, where θ is the angle between them such that 0≤θ≤π.

What is cross product?

o It is denoted by A x B by placing a

cross sign between the vectors. o So we have the equation,

A x B = AB sinθη = Co Another name of dot product is vector product.

History of dot product:

Dot product was founded in 1901 in Vector Analysis by Edwin Bidwell Wilson:

“ The direct product is denoted by writing the two vectors with a dot between them as A.B ”

“ This is read A dot B and therefore may often be called the dot product instead of the direct product ”

History of cross product:

o The first traceable work on ”cross product” was founded in the book Vector Analysis.

o It was founded upon the lectures of Josiah Willard Gibbs, second edition by Edwin Bidwell Wilson published in 1909.

History of cross product:

o On page 61, the mention of cross product was found for the first time.

“ The skew product is denoted by a cross as the direct product was by a dot. It is written C = A x B and read A cross B. For this reason it is often called the cross product ” – Vector Analysis

Developing to present:

While studying vector analysis, Gibbs noted that the product of quaternions always had to be separated into two parts:

1. One dimensional quantity2. A three dimensional vector

Developing to present:

To avoid this complexity he proposed definingdistinct dot and cross products for pair of vectorsand introduced the now common notation for them.

Confusion about representation:

Dot product : Tait : Sαβ =Sβα

Gibbs : α.β = β.α Cross product :

Tait : Vαβ = –Vβα Gibbs : α x β = –β x α

To avoid this representation complexity, Gibbs’ notation is used universally.

Illustration of dot product:

o Why Dot Product?

-To express the angular relationship between two

vectors.

Illustration of dot product:

If A and B are two vectors of form,A = A1i + A2j +A3kB = B1i + B2j + B3k

Then the dot product of A and B is,A.B = A1B1 + A2B2 + A3B3

Illustration of dot product:

The angular relationship of two vectors A and B as per dot product is:

A.B = A B cosθ = AB cosθ

Illustration of dot product:

The dot relationship of unit vectors along three axes :

i . j = j . k = k . i = 0 and i . i = j . j = k . k = 1

Illustration of cross product:

o Why Cross Product?

- For accumulation of interactions between different

dimensions.

Illustration of cross product:

If A and B are two vectors of formA = A1i + A2j +A3kB = B1i + B2j + B3k

Then the cross Product of A and B is,

A x B =

i j kA1 A2

A3

B1 B2

B3

Illustration of cross product:

The angular relationship of two vectors A and B is

A x B = A B sinθ = AB sinθ

Illustration of cross product:

The cross relationship of unit vectors along three axes are:

i x i = j x j = k x k = 0i x j = k & j x i = -k

j x k = i & k x j = -i k x i = j & i x k = -j

Dot product vs cross product:Dot product Cross product

Result of a dot product is a scalar quantity.

Result of a cross product is a vector quantity.

It follows commutative law. It doesn’t follow commutative law.

Dot product of vectors in the same direction is maximum.

Cross product of vectors in same direction is zero.

Dot product of orthogonal vectors is zero.

Cross product of orthogonal vectors is maximum.

It doesn’t follow right hand system.

It follows right hand system.

It is used to find projection of vectors.

It is used to find a third vector.

It is represented by a dot (.)

It is represented by a cross (x)

Properties of dot product:

☻ Commutative law: A.B = B.A☻ Distributive law: A.(B+C) = A.B+A.C

☻ Associative law: m(A.B) = (mA).B = A.(mB)

Properties of cross product:

☻ Distributive law: A x (B+C) = A x B+A x C

☻ Associative law: m(A x B) = (mA) x B = A x (mB)

Distinction in commutative law:

A x B = C has a magnitude ABsin and direction is such that A, B and C form a right handed system (from fig-a )

θ

A x B = C

A BFig - (a)

Distinction in commutative law:

B x A = D has magnitude BAsin anddirection such that B, A and D form aright handed system ( from fig -b )

B x A = D

Fig - (b)

A B

Distinction in commutative law:

Then D has the same magnitude as C but is opposite in direction,

that is, C = - D A x B = - B x A

Therefore the commutative law for cross product is not valid.

Applications of dot product:

❶ Finding angle between two vectors:

A.B = |A||B| coscos = = )

A

B

❷ Projections of light:

B

A

Light source

NO

cos = ON = B cos

From the figure,

cos = B cos =

As we know,

ON = So we reach to,

Real life applications of dot product:

o Calculating total costo Electromagnetism, from which we get

light, electricity, computers etc.o Gives the combined effect of the

coordinates in different dimensions on each other.

Applications of cross product:

❶ To find the area of a parallelogram:

Area of parallelogram = h |B| = |A| sinθ |B| = | A x B |

A

B

h

O

C

Applications of cross product:

❷ To find the area of a triangle:

Area of triangle = h |B| = |A| sinθ |B| = | A x B |

A

B

h

O

Real life applications of cross product:

o Finding momento Finding torqueo Rowing a boato Finding the most effective path

Dot and cross vector together:

Dot and cross products of three vectors A , B and C mayproduce meaningful products of the form (A.B)C, A.(BxC)and Ax(BxC) then phenomenon is called triple product.

A.(B x C) =

A1 A2 A3

B1 B2 B3

C1 C2 C3

Application of triple product:

hn

A

BC

Volume of the parallelepiped

= (height h) x (area of the parallelogram I)

= (A.n) x (| B x C |) = A. (| B x C | n) = A. ( B x C )

I

Memory booster:

Area of a triangle of vectors is determined by which

vector product method?A. Dot B. Cross

Memory booster:

Area of a triangle of vectors is determined by which

vector product method?A. Dot B. Cross

Memory booster:

Area of a triangle of vectors is determined by which

vector product method?A. Dot B. Cross

Projection of vectors is determined by which vector product method?

A. Cross B. Dot

Memory booster:

Area of a triangle of vectors is determined by which

vector product method?A. Dot B. Cross

Projection of vectors is determined by which vector product method?

A. Cross B. Dot

Thank you