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XYLENE POWER LTD.

VECTOR IDENTITIES

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

The following vector identities are used at various places on this web site.

UNIT VECTOR DOT PRODUCT PROPERTIES:
Unit vectors x, y, z are orthogonal if:
x * y = y * x = 0
x * z = z * x = 0
y * z = z * y = 0
and are also 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
 

UNIT VECTOR CROSS PRODUCT PROPERTIES:
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

This mathematical identity is used for decomposition of momentum.
 

This web page last updated August 29, 2013.

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