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Let *U* = Ux* i* + Uy* j *+ Uz **k** ,

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

let* V *= Vx **i +** Vy **j** + Vz **k**

Now,*U* . **V** = Ux Vx + Uy Vy + Uz Vz = dot product

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

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

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

(*U* Χ **V**) Χ *V*

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

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

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

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

+ 2 Ux Vx Uz Vz)

(*U* . *V*)² - [ (*U* Χ **V**) Χ *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)

= |*U* |² * | *V* |²

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

========================================

A simpler method.

We use the* Triple product expansion or Lagrange's formula *for vector products:

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

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

So [ (*U* Χ **V**) Χ *V *] . *U*

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

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

= (*V* . *U*)² - | *V* |² * | *U* |²

Hence,

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

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

= |*V* |² * | *U* |²

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,

(

= (

= |