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Let

i, j and k are unit vectors in x, y and z directions respectively.

let

Now,

= Ux Vy

= (Uy Vz - Uz Vy)

(

= (Uy Vz - Uz Vy)Vy

+ (Uz Vx - Ux Vz) Vz

= (Uy Vy Vz + Ux Vx Vz - Uz Vy² - Uz Vx² )

+ (Uz Vy Vz + Ux Vx Vy - Uy Vz² - Uy Vx²)

+ (Uz Vx Vz + Uy Vx Vy - Ux Vz² - Ux Vy² )

[ (

= (Ux Vx Uz Vz + Ux Vx Uy Vy - Ux² Vz² - Ux² Vy²)

+ (Uy Uz Vy Vz + Ux Uy Vx Vy - Uy² Vz² - Uy² Vx²)

+ (Uy UzVy Vz + Ux Uz Vx Vz - Uz² Vy² - Uz² Vx²)

(

+ 2 Ux Vx Uz Vz)

(

= Ux² Vx² + Uy² Vy² + Uz² Vz² + Ux² Vz² + Ux² Vy² + Uy² Vz² + Uy² Vx²

+ Uz² Vy² + Uz² Vx² --- other terms cancel each other.

= Ux² (Vx² +Vy²+Vz²) + Uy² (Vx² +Vy² + Vz²) + Uz² (Vx² + Vy² + Vz²)

= (Ux² + Uy² + Uz²) (Vx² + Vy² + Vz²)

= |

This is a long and detailed method.. a little difficult.

A simpler method.

We use the

(

So [ (

= [ (

= (

= (

Hence,

(

= (

= |

**= Ux***U***+ Uy***i***+ Uz***j**,***k**i, j and k are unit vectors in x, y and z directions respectively.

let

**= Vx***V**Vy***i +***+ Vz***j****k**Now,

**.***U**= Ux Vx + Uy Vy + Uz Vz = dot product***V***U Χ***cross product = (Ux***V =**+ Uy***i***+ Uz***j****) X (Vx***k***+ Vy***i**+ Vz***j****)***k*= Ux Vy

*- Ux Vz***k***- Uy Vx***j***+ Uy Vz***k***+ UzVx***i***- Uz Vy***j***i*= (Uy Vz - Uz Vy)

**+ (Uz Vx - Ux Vz)***i**+ ((Ux Vy - Uy Vx)***j****k**(

**Χ***U**) Χ***V****[ (Uy Vz-Uz Vy)***V*

==

**+ (Uz Vx-Ux Vz)***i**+ (Ux Vy-Uy Vx)***j****k] Χ**(Vx*Vy***i +***+ Vz***j***)***k**= (Uy Vz - Uz Vy)Vy

*- (Uy Vz - Uz Vy) Vz***k***- (Uz Vx - Ux Vz) Vx***j***k*+ (Uz Vx - Ux Vz) Vz

*+ (Ux Vy - Uy Vx) Vx***i***- (Ux Vy - Uy Vx)***j****Vy***i*= (Uy Vy Vz + Ux Vx Vz - Uz Vy² - Uz Vx² )

**k**+ (Uz Vy Vz + Ux Vx Vy - Uy Vz² - Uy Vx²)

**j**+ (Uz Vx Vz + Uy Vx Vy - Ux Vz² - Ux Vy² )

**i**[ (

**Χ***U**) Χ***V****=***V ] .*U= (Ux Vx Uz Vz + Ux Vx Uy Vy - Ux² Vz² - Ux² Vy²)

+ (Uy Uz Vy Vz + Ux Uy Vx Vy - Uy² Vz² - Uy² Vx²)

+ (Uy UzVy Vz + Ux Uz Vx Vz - Uz² Vy² - Uz² Vx²)

(

**.***U***)² = (Ux² Vx² + Uy² Vy² + Uz² Vz² + 2 Ux Vx Uy Vy + 2 Uy Vy Uz Vz***V*+ 2 Ux Vx Uz Vz)

(

**.***U***)² - [ (***V***Χ***U**) Χ***V****=***V ] .*U= Ux² Vx² + Uy² Vy² + Uz² Vz² + Ux² Vz² + Ux² Vy² + Uy² Vz² + Uy² Vx²

+ Uz² Vy² + Uz² Vx² --- other terms cancel each other.

= Ux² (Vx² +Vy²+Vz²) + Uy² (Vx² +Vy² + Vz²) + Uz² (Vx² + Vy² + Vz²)

= (Ux² + Uy² + Uz²) (Vx² + Vy² + Vz²)

= |

**|² * |***U***|²***V*This is a long and detailed method.. a little difficult.

**========================================**

A simpler method.

We use the

**for vector products:***Triple product expansion or Lagrange's formula***(***A X***) = (***B X C***.***C***)***A***(***B -**.***A****)***B**C*(

**X***A***)***B**X***= (***C***.***C***)***A**B***(***-***.***B***) A***C*So [ (

**Χ***U**) Χ***V****] .***V**U*= [ (

**.***V***)***U***- (***V***.***V***)***V***] .***U**U*= (

**.***V***) (***U***.***V***) - (***U***.***V***) (***V***.***U***)***U*= (

**.***V***)² - |***U***|² * |***V***|²***U*Hence,

(

**.***U***)² - [ (***V***Χ***U**) Χ***V***V ] .*U= (

**.***U***)² - [ (***V***.***V***)² - |***U***|² * |***V***|² ]***U*= |

**|² * |***V***|²***U*