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XYLENE POWER LTD.

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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