Summary of Vectors

Summary of Vector Properties
Basics Quantities - Scalars and Vectors
There are quantities, such as velocity, that have both magnitude and direction. These are
vectors. A quantity with only magnitude is a scalar.
Nomenclature
r r r r
A,B,C,D
A r
ê
x
,ê , ê
y
z
A x ,A y ,A z
s
will be vectors
is the magnitude or length of the vector A r . It is a scalar.
will be a set of orthogonal unit vectors along the x,y,z axes
(Some authors use i,j,k for these unit vectors.)
are scalars that are the components of the vector A r
is a scalar
Component Notation
r
A =
=
A
x
(A
ê
x
x
+ A
, A
y
y
ê
, A
y
z
+ A
Scalar Multiplication
r
s A = s (A
x
, A
y
, A
z
) =
r r r r
s (A + B) = sA + s B
=
)
z
ê
z
(sA
x
, sA , sA )
(sA x + sB x , sA y + sB y , sA z + sB
y
z
z
)
Addition
r r
A + B
=
=
r r
B + A
(A x + B x , A y + B y,
A z + B
z
)
Dot Product or Inner Product
A
r • B
r
is a scalar.
r r r r
A • B = A B cosθ
= A xB
x
+ A yBy
+ A zBz
r
Where θ is the angle between A and
r r
A • B =
r r
B • A
r
B
. (Either angle can be used.)
1

A • (B + C)
r r
s(A • B) =
r r r r r r r
= A • B + A • C = (B + C) • A
r r r r
(sA) • B = A • (sB)
Magnitude or Length
r
A
=
r r
A • A
=
A
2
x
+ A
2
y
+ A
2
z
Cross Product or Outer Product
r
D
0
D r
r r
r r
= A × B is a vector (pseudo-vector). It is perpendicular to both A and B
r r r r
= D • A = D • B
= A B sin θ
The direction of D r is perpendicular to
r
A
and
advance when A r is rotated into B r . (Right hand rule.)
r
B , in the direction of a right hand screw
The cross product is not commutative,
r r
A × B
=
r r
− B × A
r
D
=
=
r r
A × B
( A
y
B
z
− A
z
B
y
,
A
y
B
z
− A
z
B
y
,
A
y
B
z
− A
z
B
y
)
Note the cyclic nature of the multiplications in component notation.
Formally, manipulating symbols, we can represent the cross product as a determinate.
r
D
=
=
r r
A × B
det
ê
A
B
x
x
x
ê
A
B
y
y
y
Orthogonal Unit (Orthonormal) Vectors
From the properties of the inner product one has:
ê
A
B
z
z
z
2

3
0
ê
ê
ê
ê
ê
ê
1
ê
ê
ê
ê
ê
ê
z
y
z
x
y
x
z
z
y
y
x
x
=
•
=
•
=
•
=
•
=
•
=
•
The magnitude of unit vectors cross products are:
0
ê
ê
ê
ê
ê
ê
1
ê
ê
ê
ê
ê
ê
z
z
y
y
x
x
x
z
z
y
y
x
=
×
=
×
=
×
=
×
=
×
=
×
The cross products are:
)
ê
ê
(
1
ê
ê
)
ê
ê
(
1
ê
ê
)
ê
ê
(
1
ê
ê
z
x
x
z
y
z
z
y
x
y
y
x
×
−
=
=
×
×
−
=
=
×
×
−
=
=
×