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 cosθ
o 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 = C
o Another name of dot product is
vector product.

History of dot product:
Dot product was founded in 1901 in Vector Analysis by Edwin
BidwellWilson:
“ 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 quantity
2. A three dimensional vector

Developing to present:
To avoid this complexity he proposed defining
distinct dot and cross products for pair of vectors
and 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.

If A and B are two vectors of form,
A = A1i + A2j +A3k
B = B1i + B2j + B3k
Then the dot product of A and B is,
A.B = A1B1 + A2B2 + A3B3

The angular relationship of two vectors A and B as
per dot product is:
A.B = A B cosθ
= AB cosθ

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.

If A and B are two vectors of form
A = A1i + A2j +A3k
B = B1i + B2j + B3k
Then the cross Product of A and B is,
A x B =
i j k
A1 A2 A3
B1 B2 B3

The angular relationship of two vectors A and B is
A x B = A B sinθ
= AB sinθ

The cross relationship of unit vectors along three
axes are:
i x i = j x j = k x k = 0
i 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 magnitudeABsin and
direction is such that A, B and C form a right
handed system (from fig-a ) θ
A x B = C
A B
Fig - (a)

B x A = D has magnitude BAsin and
direction such that B, A and D form a
right handed system ( from fig -b )
B x A = D
Fig - (b)
A B

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| cos
cos =
A.B
AB
 = cos−1
(
A.B
AB
)

A
B

❷ Projections of light:
B
A

Light source
NO
cos =
ON
B
ON = B cos
From the figure,
cos =
A.B
AB
B cos =
A.B
A
As we know,
ON =
A.B
A
So we reach to,

Real life applications of dot product:
o Calculating total cost
o 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 =
1
2
h |B|
=
1
2
|A| sinθ |B|
=
1
2
| A x B |

A
B
h
O

Real life applications of cross product:
o Finding moment
o Finding torque
o Rowing a boat
o Finding the most effective path

Dot and cross vector together:
Dot and cross products of three vectors A , B and C may
produce 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:
h
n
A
B
C
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
 Projection of vectors is determined by which vector
product method?
A. Cross B. Dot

Dot & cross product of vectors

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