Synthesis
3 Communications

7 The Dyadic Product Derivation

Screen

  1 Curiosity,
  2 Purpose,
  3 Materials,
  4 Apparatus,
  5 Procedure,
  6 Observations,
  7 Thesis,
  8 Analysis,
  9 Conclusions,
10 Discussion.
11 Quaternions










































Curiosity:

3.7 Screen 1

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Given two vectors, what is the general product?








































Purpose:

3.7 Screen 2

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The purpose of this activity is to derive the dyadic product (general product) of two vectors and show its application to area (the vector area of a parallelogram defined by two vectors) and volume (the scalar volume of a parallelepiped defined by three vectors).  

We define area in terms of two displacements describing adjacent sides of a parallelogram. To see the a students investigation of the general dyadic product go to Student Dyadic










































Materials:

3.7 Screen 3

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Given two vectors the area  defined by R11 and R22:

4.2.0 R 11 R1x x + R1y y + R1zz and  R 22 R2x x + R2y y + R2zz.








































Apparatus Diagram:

3.7 Screen 4

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From 5.2.2 Diagram 2: Two points in the room.











































Procedure (observations):

3.7 Screen 5

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1) Given two vectors in a Cartesian system..
2) Derive the general dyadic product.
3) Define the unit vector rules.
4) Define the dot and cross product parts of the dyadic product..







































Observations:

3.7 Experiment 1 Screen 6
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3.7 Equation 1 r1r1 = r1xx + r1yy + r1zz.

3.7 Equation  2r2r2 = r2xx + r2yy + r2zz.









































Thesis:

3.7 Screen 7

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From 5.2.0 Space - The First Of The Three Basic Postulates Of Physical Reality:

5.2.0  R 00 R0x x + R0y y + R0zz.

Given  R11 and R22:

5.2.0 R 11 R1x x + R1y y + R1zz and  R 22 R2x x + R2y y + R2zz.

We define the general dyadic product D1212 of two vectors  R 11  and  R 22 :

3.7 Definition 1D1212  + R1xR2xxx +  R1yR2yyy + R1zR2zzz
  + R1xR2yxy + R1xR2zxz
+ R1yR2xyx + R1yR2zyz
+ R1zR2xzx + R1zR2yzy .


3.7 Definitions 2 The unit vector products (by using the right hand rule):

xx   1,  xy  zxz -y
yx - zyy 1,   yz x
zx    yzy -x, zz 1, 

Using 3.7 Definitions 2 The unit vector products in 3.7 Definition  1 gives:

3.7 Equation 3(R1R1)( R2R2) = R1xR2xxx +  R1y R2yyy + R1z R2zzz
+ R1xR2yxy + R1xR2zxz
+ R1y R2xyx + R1y R2zyz
+ R1z R2xzx + R1z R2yzy

=  R1xR2x +  R1y R2y +R1z R2z
+ R1xR2yz -  R1xR2z y
- R1y R2xz + R1y R2zx
+ R1z R2xy - R1z R2yx 

=  R1xR2x +  R1y R2y +R1z R2z
+ (R1y R2z- R1z R2y) x  
+ ( R1z R2x-  R1xR2z)
+ (R1xR2y - R1y R2x) z .

From the scalar part of 3.7 Equation 3 ( R1xR2x+ R1y R2y +R1z R2z) we define the dot product or scalar product part and with out derivation the cos q as :

3.7 Definition 3 (R1R1 R2R2 R1xR2x +  R1y R2y +R1z R2z,
= |R1| | R2| cos q12 .

From the vector part of 3.7 Equation 3 ( (R1yR2z- R1zR2y)x+( R1zR2x-  R1xR2z)y+(R1xR2y- R1yR2x)z ) we define the cross product or vector product part as:

3.7 Definition 4 (R1R1 R2R2 (R1y R2z- R1z R2y) x  
+ ( R1zR2x-  R1xR2z)
+ (R1xR2y - R1y R2x) z
= |R1| | R2| sin q12 12

Using 3.7 Definition 3 (R1R1 R2R2) and 3.7 Definition 4 (R1R1 R2R2) in  the Dyadic Product 3.7 Equation 3 (R1R1)( R2R2) = gives:

3.7 Equation 4(R1 R1)( R2 R2) = (R1 R1  R2 R2) + (R1 R1  R2 R2),
= |R1| | R2| sin q12 12 + |R1| | R2| cos q12






































Analysis (observations):

3.7 Screen 8

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Given two vectors defined by R11 and R22:

4.2.0 R 11 R1x x + R1y y + R1zz and  R 22 R2x x + R2y y + R2zz.

3.7 Equation 1 r1r1 = r1xx + r1yy + r1zz.

3.7 Equation  2r2r2 = r2xx + r2yy + r2zz.

Using 3.7 Equation  1 (r1r1 = r1xx + r1yy + r1zz) and 3.7 Equations  2 (r2r2 = r2xx + r2yy + r2zz) in 3.7 Definition  3 the dyadic product is defined as (r1r1)( r2r2).

3.7 Definition  1(r1r1)( r2r2 (r2xx + r2yy + r2zz)( r1xx + r1yy + r1zz),
=  + r1xxr2xx + r1xxr2yy + r1xxr2zz
+ r1yy r2xx + r1yy r2yy + r1yy r2zz
+ r1zz r2xx + r1zz r2yy + r1zz r2zz


3.7 Definitions 2 The unit vector products (by using the right hand rule):

xx   1,  xy  zxz -y
yx - zyy 1,   yz x
zx    yzy -x, zz 1, 

3.7 Equation 3(r1r1)( r2r2) =   r1xr2x +  r1y r2y +r1z r2z
+ (r1y r2z- r1z r2y) x  
+ ( r1z r2x-  r1xr2z)
+ (r1xr2y - r1y r2x) z .

From the vector part of 3.7 Equation 3 ( = (r1yr2z- r1zr2y)x+( r1zr2x-  r1xr2z)y+(r1xr2y- r1yr2x)z ) we define the cross product or vector product part as:

3.7 Definition 4 (r1r1 r2r2 (r1y r2z- r1z r2y) x  
+ ( r1zr2x-  r1xr2z)
+ (r1xr2y - r1y r2x) z
= |r1| | r2| sin q12 12

Using 3.7 Definition 3 (r1r1 r2r2) and 3.7 Definition 4 (r1r1 r2r2) in  the Dyadic Product 3.7 Equation 3 (r1r1)( r2r2) = gives:

3.7 Equation 4(r1 r1)( r2 r2) = (r1 r1  r2 r2) + (r1 r1  r2 r2),
= |r1| | r2| sin q12 12 + |r1| | r2| cos q12


































Conclusions:

3.7 Experiment 1 Screen 9
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Using 3.7 Equation  1 (r1r1 = r1xx + r1yy + r1zz) and 3.7 Equations  2 (r2r2 = r2xx + r2yy + r2zz) in 3.7 Definition  3 the dyadic product is defined as (r1r1)( r2r2).

3.7 Definition  1(r1r1)( r2r2 (r2xx + r2yy + r2zz)( r1xx + r1yy + r1zz),
=  + r1xxr2xx + r1xxr2yy + r1xxr2zz
+ r1yy r2xx + r1yy r2yy + r1yy r2zz
+ r1zz r2xx + r1zz r2yy + r1zz r2zz


3.7 Definitions 2 The unit vector products (by using the right hand rule):

xx   1,  xy  zxz -y
yx - zyy 1,   yz x
zx    yzy -x, zz 1, 

3.7 Equation 3(r1r1)( r2r2) =   r1xr2x +  r1y r2y +r1z r2z
+ (r1y r2z- r1z r2y) x  
+ ( r1z r2x-  r1xr2z)
+ (r1xr2y - r1y r2x) z .

From the vector part of 3.7 Equation 3 ( = (r1yr2z- r1zr2y)x+( r1zr2x-  r1xr2z)y+(r1xr2y- r1yr2x)z ) we define the cross product or vector product part as:

3.7 Definition 4 (r1r1 r2r2 (r1y r2z- r1z r2y) x  
+ ( r1zr2x-  r1xr2z)
+ (r1xr2y - r1y r2x) z
= |r1| | r2| sin q12 12

Using 3.7 Definition 3 (r1r1 r2r2) and 3.7 Definition 4 (r1r1 r2r2) in  the Dyadic Product 3.7 Equation 3 (r1r1)( r2r2) = gives:

3.7 Equation 4(r1 r1)( r2 r2) = (r1 r1  r2 r2) + (r1 r1  r2 r2),
= |r1| | r2| sin q12 12 + |r1| | r2| cos q12










































Discussion:

3.7 Experiment 1 Screen 10
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This chapter is communications. We will use and prove these language skills in applications later on. 









































End