Download Summary of Vector Properties Basics Quantities

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
r
A
will be vectors
r
is the magnitude or length of the vector A . It is a scalar.
ê x ,ê y , ê z
will be a set of orthogonal unit vectors along the x,y,z axes
(Some authors use i,j,k for these unit vectors.)
r
are scalars that are the components of the vector A
is a scalar
Ax,Ay,Az
s
Component Notation
r
A = A x ê x + A y ê y + A z ê z
= (A x , A y , A z )
Scalar Multiplication
r
s A = s (A x , A y , A z ) = (sA x , sA y , sA z )
r r
r
r
s (A + B) = s A + s B
= (sA x + sB x , sA y + sB y , sA z + sB 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
r r
A • B is a scalar.
r r
r r
A • B = A B cos θ
= A x B x + A y By + A z Bz
r
r
Where θ is the angle between A and B . (Either angle can be used.)
r r
r r
A• B = B• A
1
r r r
r r r r
r r r
A • ( B + C) = A • B + A • C = ( B + C) • A
r r
r r
r
r
s(A • B) = (sA ) • B = A • (sB)
Magnitude or Length
r
A
=
r r
A•A
=
A 2x + A 2y + A 2z
Cross Product or Outer Product
r
r r
r
r
D = A × B is a vector (pseudo-vector). It is perpendicular to both A and B
r r
r r
0 = D•A = D•B
r
D = A B sin θ
r
r
r
The direction of D is perpendicular to A and B , in the direction of a right hand screw
r
r
advance when A is rotated into B . (Right hand rule.)
The cross product is not commutative,
r r
r r
A×B = − B×A
r r
r
D = A×B
= ( A y Bz − A z B y , A y Bz − A z By , A yB z − A z By )
Note the cyclic nature of the multiplications in component notation.
Formally, manipulating symbols, we can represent the cross product as a determinate.
r
r r
D = A×B
ê x
ê y
ê z
= det A x
Bx
Ay
By
Az
Bz
Orthogonal Unit (Orthonormal) Vectors
From the properties of the inner product one has:
2
ê x • ê x
= ê y • ê y
= ê z • ê z
= 1
ê x • ê y
= ê x • ê z
= ê y • ê z
= 0
The magnitude of unit vectors cross products are:
ê x × ê y
=
ê y × ê z
=
ê z × ê x
= 1
ê x × ê x
=
ê y × ê y
=
ê z × ê z
= 0
The cross products are:
ê x × ê y
= 1 = − ( ê y × ê x )
ê y × ê z
= 1 = − ( ê z × ê y )
ê z × ê x
= 1 = − ( ê x × ê z )
3