Casio fx-4000P: Orthonormal 2D Vector Test

Casio fx-4000P:  Orthonormal 2D Vector Test

Introduction

A set of vectors form an orthonormal basis if the vectors are mutually perpendicular and are of unit length.  

Mutual perpendicular means the dot product of two different vectors in the space is 0:

v_i ∙ v_j = 0,  i ≠ j

Unit length means that the norm of each vector is 1:

|v_i| = 1

 For two 2D vectors [ A, B ] and [ C, D ], the two vectors form an orthonormal basis if:

A^2 + B^2 = 1

C^2 + D^2 = 1

A * D + B * C = 0

Casio fx-4000P Program:  Orthonormal 2D Vector Test

Bytes:  84 steps

(line returns are included for readability)

"A":

?→A:

"B":

?→B:

"C":

?→C:

"D":

?→D:

A²+B²≠1⇒Goto 0:

C²+D²≠1⇒Goto 0:

AD+BC≠0⇒Goto 0:

"YES" ⊿

Goto 1:

Lbl 0:

"NO" ⊿

Lbl 1

Examples

Example 1:

[1, 0], [0, 1]

A = 1, B = 0, C = 0, D = 1

"NO"   (1*1 - 0*0 = 1)

Example 2:

[1/√2, 1/√2], [1/√2, -1/√2]

A = 1/√2, B = 1/√2, C = 1/√2, D = -1/√2

"YES"

Source:

Rowland, Todd. "Orthonormal Basis." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/OrthonormalBasis.html

Last Accessed April 7, 2022

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Applications: Sharp EL-5150

Applications:  Sharp EL-5150 

Note:  spaces included for readability

Fan Laws

AER Equation:

1; f(BCDI) = B * ( C ÷ D ) Y^x ( 1 ÷ I  )

Variables:

Calculate CPM_new

B = CPM_old

1st Fan Law:

I = 1

C = RPM_new

D = RPM_old

2nd Fan Law:

I = 2

C = SP_new

D = SP_old

3rd Fan Law:

I = 3

C = BHP_new

D = BHP_old

Example 1:

Fan Law 2:

CPM_old = B = 4000 CPM

SP_new = C = 48 

SP_old = D = 36

I = 2

Result:  4618.802153

Example 2:

Fan Law 3:

CPM_old = B = 3500 CPM

BHP_new = C = 59

BHP_old = D = 52

I = 3

Result:  3650.488072

Ideal Shockley Diode Equation

I = I0 * e^((VD/(n* VT) - 1)

where VT = K * T/q

I = diode current (amps)

I0 = saturation current (amps)

VT = thermal voltage (V) - see notes below

VD = voltage across the diode (V)

n = ideality factor, in ideal situations, n = 1

Notes:  

*  The equation below assumes the ideal diode, n = 1

*  The equation uses a ratio of scientific constants:  k/q 

*  k = Boltzmann's Constant = 1.380649 * 10^-23 J/K

*  q = Charge of an Electron = 1.602176634 * 10^-19 C  (on some calculators, like the Casio fx-991EX, this constant is labeled e)

*  k/q = 8.617332385 * 10^-5 J/(K*C) = 8.617332385 * 10^-5 V/K  (volts/degrees Kelvin)

AER Equation:

1; f(IDE) = 8.617332385E-5 × E STO A, I ×(e(D ÷ A) - 1)

Calculate VT (stored in A), the I 

I = I0

D = VD

E = temperature in Kelvin

Example 1:

I = 4E-6 A 

D = 0.08 V

E = 280 K

Results:

A = 0.024128531, (I) 0.00001061

 

Example 2:

I = 4E-6 A 

D = 0.06 V

E = 300 K

Results:

A = 0.025852000, (I) 0.00003674

Dot and Cross Product of Two 3D Vectors

For the two vectors [A, B ,C] and [D, E, F]:

AER Equations:

1; f(ABCDEF) = A × D + B × E + C × F ◣

2; B × F - C × E, C × D - A × F, A × E - B × D

Example 1:

[ A, B, C ] and [ D, E, F]

[ 4.5, -2.5, -8 ] and [ 1.6, 3.9, 6 ]

Dot Product:  -50.55

Cross Product:  [ 16.2, -39.8, 21.55 ]

Example 2:

[ A, B, C ] and [ D, E, F]

[ 4, 3, 2 ] and [ 2, 7, 0 ]

Dot Product:  29

Cross Product:  [ -14, 4, 22 ]

Law of Cosines

Sides with lengths A, B, C with D as the angle opposite of A.  Equation 1 finds the length of side A, while Equation 2 finds the angle D.

AER Equations:

1; f(BCD) = √(B^2 + C^2 - 2 × B × C × COS D) STO A ◣

2; f(ABC) = cos^-1 ((B^2 + C^2 - A^2) ÷ (2 × B × C)) STO D

Example 1 - find A:

Degree Mode Set

Input:  B = 4.5, C = 3.7, D = 30°

Run 1:

Result:  2.258617731

Example 2 - find D:

Degree Mode Set

Input:  A = 40, B = 56, C = 38

Run 2:

Result:  45.5579132°

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Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.