XYLENE POWER LTD.

VECTOR IDENTITIES

By Charles Rhodes, P.Eng., Ph.D.

VECTORS:
Vectors are mathematical representations of physical parameters which inherently have both direction and magnitude. Examples of parameters that are frequently represented by vectors are position, velocity and electric, magnetic and gravitational field strength. Vector representations tend to scare novices but they actually enable simplification of multi-dimensional problems.
 

VECTOR FORM:
Typically a vector A takes the form:
A = Ax x + Ay y + Az z
where:
x = unit vector parallel to x axis;
y = unit vector parallel to y axis;
z = unit vector parallel to z axis;
Ax = Vector A's magnitude component parallel to x axis;
Ay = Vector A's magnitude component parallel to y axis;
Az = Vector A's magnitude component parallel to z axis;
 

VECTOR MAGNITUDE:
The magnitude |A| of any vector A is given by:
|A|^2 = Ax^2 + Ay^2 + Az^2
This formula is simply a generalization of Pythagorus theorm.
 

UNIT VECTOR DOT PRODUCT PROPERTIES:
By definition unit vectors x, y, z are orthogonal if:
x * y = y * x = 0
x * z = z * x = 0
y * z = z * y = 0
and by definition unit vectors x, y, z are orthonormal if:
x * x = 1
y * y = 1
z * z = 1
 

VECTOR DOT PRODUCT:
Consider two three dimensional vectors A and B where:
A = (Ax x + Ay y + Az z)
and
B = (Bx x + By y + Bz z)

The vector dot product A * B is given by:
A * B = (Ax x + Ay y + Az z) * (Bx x + By y + Bz z)
= Ax Bx x * x + Ay By y * y + Az Bz z * z
+ Ax By x * y + Ax Bz x * z + Ay Bx y * x
+ Ay Bz y * z + Az Bx z * x + Az By z * y
= Ax Bx + Ay By + Az Bz

A * A = |A|^2
= Ax^2 + Ay^2 + Az^2
 

GENERAL UNIT VECTOR:
A unit vector parallel to any vector A is:
[A / |A|]

Note that:
[A / |A|] * [A / |A|]
= |A|^2 / |A|^2
= 1
 

VECTOR COMPONENT:
An arbitrary vector B at any particular position can be expressed as the sum of a component parallel to vector A at the same position and a component normal to vector A at that same position. At this position the component of vector B parallel to vector A is:
(A * B) / |A|
where:
(A * B) = Ax Bx + Ay By + Az Bz

Hence:
A * B) / |A|
= (Ax Bx + Ay By + Az Bz) / (Ax^2 + Ay^2 + Az^2)^0.5
= |B| cos(Theta)
where (Theta) is the angle between vectors A and B.
 

UNIT VECTOR CROSS PRODUCT:
Orthonormal unit vector cross product properties are:
x X y = z
y X z = x
z X x = y
y X x = -z
z X y = -x
x X z = -y
x X x = 0
y X y = 0
z X z = 0
 

VECTOR CROSS PRODUCT:
Consider the vector cross product (A X B) where:
A = Ax x + Ay y + Az z
and
B = Bx x + By y + Bz z

(A X B) = (Ax x) X (Bx x + By y + Bz z)
+ (Ay y) X (Bx x + By y + Bz z)
+ (Az z) X (Bx x + By y + Bz z)
 
= +Ax Bx (x X x) + Ax By (x X y) + Ax Bz (x X z)
+Ay Bx (y X x) + Ay By (y X y) + Ay Bz (y X z)
+Az Bx (z X x) + Az By (z X y) + Az Bz (z X z)
 
= + Ax By (z) + Ax Bz (-y)
+Ay Bx (-z) + Ay Bz (x)
+Az Bx (y) + Az By (-x)
 
= + (Ay Bz – Az By) x + (Az Bx – Ax Bz) y + (Ax By - Ay Bx) z
or
(A X B)
= + (Ay Bz – Az By) x + (Az Bx – Ax Bz) y + (Ax By -Ay Bx)z
 

VECTOR IDENTITY:
Recall that:
(A X B) = + (Ay Bz – Az By) x + (Az Bx – Ax Bz) y + (Ax By -Ay Bx) z
Hence:
(A X B) * (A X B)
= |A * B|^2
= (Ay Bz – Az By)^2 + (Az Bx – Ax Bz)^2 + (Ax By -Ay Bx)^2

Recall that:
(A * B) = Ax Bx + Ay By + Az Bz

Hence:
|A * B|^2 = (A * B)^2
= (Ax Bx + Ay By + Az Bz) (Ax Bx + Ay By + Az Bz)
= Ax^2 Bx^2 + Ay^2 By^2 + Az^2 Bz^2 + 2 Ax Bx Ay By + 2 Ax Bx Az Bz + 2 Ay By Az Bz

Thus:
(A X B) * (A X B) + (A * B)^2
 
= (Ay Bz – Az By)^2 + (Az Bx – Ax Bz)^2 + (Ax By -Ay Bx)^2 + Ax^2 Bx^2
+ Ay^2 By^2 + Az^2 Bz^2 + 2 Ax Bx Ay By + 2 Ax Bx Az Bz + 2 Ay By Az Bz
 
= Ay^2 Bz^2 + Az^2 By^2 + Az^2 Bx^2 + Ax^2 Bz^2 + Ax^2 By^2 + Ay^2 Bx^2
+ Ax^2 Bx^2 + Ay^2 By^2 + Az^2 Bz^2
 
= (Ax^2 + Ay^2 + Az^2) (Bx^2 + By^2 + Bz^2)
 
= (A * A) (B * B)
 
= |A|^2 |B|^2

Hence we have proven the vector identity:
|A X B|^2 + (A * B)^2 = |A|^2 |B|^2

If vectors A and B share the same location then:
|A|^2 |B|^2 [cos(Theta)]^2
+ |A|^2 |B|^2 [sin(Theta)]^2
= |A|^2 |B|^2
or
[cos(Theta)]^2 + [sin(Theta)]^2 = 1
where:
(Theta) is the angle between vector A and vector B

This mathematical identity is used for decomposition of a portion of the momentum of an energy packet into radial and angular components.
 

This web page last updated September 19, 2020.