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Vectors
VECTORS
SCALARS : Physical Properties that are specified only by their
magnitude
e.g. Mass , length,energy
VECTORS : Physical properties that are specified by their length as
well as direction
e.g. Force, position, velocity
A vector A can be defined with respect to a fixed Cartesian coordinate
system
A
k
i
j
as
A = A xi + A yj + A z k
i,j,k unit vectors along the x,y,z directions
Ax,A y,A z the projections of A on i,j,k
Vectors
Addition of two vectors
A
C
B
B
A
C = A+B = B+A
A + B = A xi + A yj + A z k +B xi + B yj + B z k
A + B = (A
x+
B x)i + (A y+B y)j + (A z +B z )k
length of vector
|A| = |
A 2x  A 2y  A2x |
Vectors
Scalar product between two vectors
The dot product or scalar product between two vectors is given by



A. B = |A||B| cos
For the unit vectors in our Cartesian coordinate system
A
k
i
j
we have
i . i = j. j = k . k = 1*1* cos(0) = 1

i . j = i. k = j.k = 1 cos(2 ) = 0
Thus
A . B = (A xi + A yj +A z ) . (B xi +B yj +B z k) =
A . B = A xBx + A yBy +A z Bz
The Vector Product or Cross product
Vectors
The vector product between two vectors A and B ,written as
AX B
C
|A| |B| sin




C = AXB
is a new vector perpendicular to the plane defined by A and B and of
the length | A||B|sin 
We have for the unit vectors in our Cartesian system
A
k
i
IXj = k
jXi =-k
j
jXk = i
kXj = - i
kXi = j
iXk = -j
Vectors
Thus
AXB = (A xi + A yj +A z ) . X(B xi +B yj +B z k)
AXB = (A yBz - Az By)i + (A z Bx -A xBz )j + (A xBy - A yBx)k
The cross product
Determinant
AXB =
can also be written as
i
AXB =
j
A
x
B
x
k
Ay
By
Az
Bz
A Vector Operator
Vectors



= i
+j
+k
x
y
z
as an example the momentum operator



p= -i [i
+j
+k
]=-i 
x
y
z
The gradient of a function is  working on the function
grad g(x,y,z) =  g(x,y,z)
grad g(x,y,z) = i
g
g
g
+j
+k
x
y
z
The gradient is a vector ,an example is the force F
derived from a scalar potential
F =-  V( x,y,z)
V
V
V
F = -i
+j
+k
x
y
z