Chapter one
The Vectors
1-6 Multiplying of Vectors
(1) Scalar Product (dot product)
r
r r
r
Let A and B are two vectors. The scalar product of A , B is
r r
A . B = A B cos q
q £ 180
q
r
Where : A is the magnitude of A
r
B is the magnitude of B
r
r
q is the angle between A and B
r
B
r
A
The quantity obtained from the scalar product is a scalar quantity. For
example, the work is a scalar quantity obtained from the product of tow
vectors (displacement and the component of force in the direction of this
displacement); W = Fd cos q
r r
Now, from: A . B = A B cos q
r r
(1) if q =0 then cos(0)=1 and A . B =AB
r r
(2) if q =90 then cos(90)=0 and A . B =0
(*) Properties of Scalar Product (Dot Product)
r r r r
(a) A . B = B. A
r r
(c) A . A = A 2 (parallel vectors)
r r r
r r r r
(b) A . ( B + C ) = A . B + A . C
r
r
(d) A . (- A )= -A2 (anti parallel vectors)
Using the unit vector î , ˆj , k̂ we obtain:
î . î = i i cos(0) =(1)(1)(1) = 1 where i=i , j=j , k=k , = 1
î . ˆj = i j cos(90) = (1)(1)(0) = 0 where i.j , j.k , i.k ,= 0
Chapter one
The Vectors
r
r
1-7 The Scalar Product of A and B Analytically
r
r
A
For two vectors and B in 3D-space
r
r
A = A x î + A y ˆj + A z k̂ , B = B x î + B y ĵ + B z k̂
r r
A . B =( A x î + A y ˆj + A z k̂ ) . ( B x î + B y ĵ + B z k̂ )
= A x î . B x î + A x î . B y ĵ + A x î . B z k̂ + A y ˆj . B x î + A y ˆj . B y ĵ + A y ˆj . B z k̂
+ A z k̂ . B x î + A z k̂ . B y ĵ + A z k̂ . B z k̂
=AxBx( î . î )+AxBy( î . ˆj )+AxBz( î . k̂ )+AyBx( ˆj . î )+AyBy( ˆj . ˆj )+AyBz( ˆj . k̂ )
AzBx( k̂ . î )+AzBy( k̂ . î )+AzBz( k̂ . k̂ )
= AxBx+ AyBy+ AzBz
r r
A xB
(2) Cross Product (Vector Product)
r
B
r
r
A
The vector product for
and B vectors is define as:
r r
A x B = A B sin f
f
r
A
Where f is the angle between the two vectors. The direction of the result
r r
A x B is obtained according to the right hand rule
(**)Properties of Vector Product
r r
r r r r r r r
r r
(a) A x B = - B x A (b) C x (A + B) = C x A + C x B
x
î
Using the unit vectors î , ˆj , k̂ we obtain:
ˆj
î x î = i i sin(0) =(1)(1)(0) = 0 = j x j=k x k
î x ˆj = k̂ , ˆj x k̂ = î , k̂ x î = ˆj
k̂
z
r
r
A
1-8 The Vector Product Of and B Analytically
y
Chapter one
The Vectors
r r
A
For two vectors , B in 3D-space,where
r
r
A = A x î + A y ˆj + A z k̂ , B = B x î + B y ĵ + B z k̂
î
r r
A x B = Ax
ˆj
k̂
Ay
A z = (AyBz- AzBy) î -( AxBz- AzBx) ˆj +( AxBy- AyBx) k̂
Bx
By
Bz
1-9 Triple Product
1-9-1 Triple Scalar Produce
r
r r
For A , B ,and C , the triple scalar product is define in the form:
r r r r r r
( A x B ). C = A .( B x C )
Ax
r r r
A .( B x C ) = B x
Ay
Az
By
B z =???? (HW)
Cx
Cy
Cz
1-9-2 Triple Vector Produce
This type of produce can be calculated by
r
r r r r r r
r r
A x ( B x C ) = (A . C) B - (A . B)C
1-10 Differentiation of Vector
The differential of vectors are the same that used for scalar quantities
r
A
Let (t) is a vector as a function to the variable (t), where:
r
A (t) = î A x ( t ) + ĵ A y ( t ) + k̂ A z ( t )
Then the differential process due to (t) is:
Chapter one
r
dA x ˆ dA y
dA z
dA
= î
+j
+ k̂
dt
dt
dt
dt
r
r
For two vectors A and B ,we can write:
r
r
r r
d A
dA dB
(a)
( +B) =
+
dt
dt
dt
r
r
r dm
d
d
A
(b)
(mA) = A
+m
dt
dt
dt
r
r
r r
r dA r dB
d A
(c)
( .B) = B.
+ A.
dt
dt
dt
r
r
r r
r dB
d A
dA r A
(d)
( xB) =
x B+ x
dt
dt
dt
The Vectors