TI-84 Plus CE and TI-83 Premium CE: Vectors Program

TI-84 Plus CE and TI-83 Premium CE:  Vectors Program

The program VECTORS is made for the TI-84 Plus CE and TI-83 Premium CE.  

Introduction

The program VECTORS calculates the following of the three dimensional vectors [A] and [B]:

* Dot product of [A] and [B]

* Euclidean norm of vectors [A] and [B]

* Angle between vectors [A] and [B] in degrees

* Cross product of [A] × [B]

* Tensor product of [A] ⊗ [B]

The program uses the system matrices [A] and [B] and formats them as 3 rows, 1 column matrices.   The program changes the angle mode to Degrees.

After entering each vector, the program gives you an option to normalize the vectors.  Since the cross and tensor products do not follow the commutative property, there is an option to switch the elements of both vectors.  

Depending on the calculator used, results will be shown in either exact format (TI-83 Premium CE) or approximate format (TI-84 CE).  

Calculations Used 

In this section, each of the vectors will be followed by the program code used.  

Let [A] and [B] be the column vectors:

[A] = [[a1][a2][a3]]

[B] = [[b1][b2][b3]]

Dot Product: 

[A] • [B] = a1 * b1 + a2 * b2 + a3 * b3 = [A]ᵀ [B]

Euclidean Norm:

||[A]||₂ = √(a1² + a2² + a3²) = √([A]ᵀ [A])

||[B]||₂ = √(b1² + b2² + b3²) = √([B]ᵀ [B])

Angle Between Two Vectors:

θ = arccos(([A] • [B]) ÷ (||[A]||₂ * ||[B]||₂))

Cross Product:

[A] × [B] = [C] where:

c1 = a2 * b3 – a3 * b2

c2 = -a1 * b3 + a3 * b1

c3 = a1 * b2 – a2 * b1

The result is a 3 x 1 matrix. 

Tensor Product:

[A] ⊗ [B] = [A] [B]ᵀ

The result is a 3 x 3 matrix. 

Download 

You can download the program and the user guide here:

https://drive.google.com/file/d/1R1sHGfr8dKQDkH87QDE0HgMDWxeLNHPN/view?usp=share_link

Eddie 

All original content copyright, © 2011-2023.  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. 

TI-86: Angle Between Vectors, Vandermonde Matrix, Least Squares Algorithm, Chebyshev Polynomials (1st Kind)

TI-86: Angle Between Vectors, Vandermonde Matrix, Least Squares Algorithm, Chebyshev Polynomials (1^st Kind)

The TI-86 has one of the best interfaces for a graphing calculator I ever had the joy to work with. The TI-86 is an update of the TI-85. Here are some programs for the TI-86.

Angle Between Vectors

The program vangle calculates the angle between two angles. The angle is calculated in degrees.

TI-86 Program vangle

(75 bytes)

Prompt V1

Prompt V2

Disp cos⁻¹ (dot(V1,V2)/(norm V1*V2))

Example: [2, -3, 4] and [8, 1, -2]

Angle: 83.5823268926°

Vandermonde Matrix

The program vander creates a matrix based on a list of coefficients.

Example: {x, y, z} produces the matrix

[ [x^0, x^1, x^2], [y^0, y^1, y^2], [z^0, z^1, z^2] ]

TI-86 Program vander

(125 bytes)

Input “List: “, L1

dimL L1 → N

{N, N} → dimL MA

For(R, 1, N)

For(C, 1, N)

L1(R)^(C-1) → MA(R,C)

End

End

Disp “MA=”

Pause MA

Example: {2, 4, 7}

Result: [ [1, 2, 4], [1, 4, 16], [1, 7, 49] ]

Least Square Algorithm

The program LSQ taxes the matrices X and Y (Y is a one-column matrix), and calculates

(X^T X)^-1 (X^T Y).

LSQ is used to fit statistical fits with least squares.

TI-86 Program LSQ

(118 bytes)

Disp “Least Squares”

Input “Matrix X: “,MX

Input “Matrix Y: “,MY

(MX^T * MX)^-1 * (MX^T * MY) → ML

Disp “ML= “

Pause ML

Example:

MX = [ [1, 3, 2.0], [1, 4, 2.3], [1, 5, 2.6], [1, 8, 2.9] ]

MY = [ [1.6], [1.8], [2.1], [2.3] ]

Results:

ML = [ [-0.14444444449], [-1.666666667E-2], [0.88888888889] ]

Chebyshev Polynomials (1^st Kind)

The program tcheby calculates the numerical value of the Chebyshev polynomials of the 1^st Kind given its point, X, and the order, N.

TI-86 Program tcheby

(127 bytes)

Prompt X,N

If X>1 : Goto A

If X<-1 :="" b="" goto="" span="">

cos(N * cos⁻¹ X) → A

Goto C

Lbl A

cosh(N * cosh⁻¹ X) → A

Goto C

Lbl B

(-1)^N * cosh(N * cosh⁻¹ X) → A

Lbl C

Disp A

Example:

X = -2.5, N = 4, Result: 263.5

X = 0.5, N = 4, Result: -0.5

X = 2.5, N = 4, Result: 263.5

Eddie

All original content copyright, © 2011-2019. 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.

Fun with the Sharp EL-5500 III (May 2018 Edition)

Fun with the Sharp EL-5500 III   (May 2018 Edition)

The EL-5500 III is one of my favorite pocket programming device, small, portable, and has my favorite programming language: BASIC. 

Let’s get started.

Note:  Substitute line numbers with labels that you find fit.  Add defined labels to the first line if you would like.  Labels can come from the bottom two rows of the QWERTY keyboard (A row and Z row).

Comments are followed by double slashes (//), they are not typed.

Sharp EL-5500 III Program:  Net Present Value

2  PAUSE “NET PRESEN VALUE”

4  CLEAR    // clears all the variables

6  INPUT “CF0:”; N, “RATE:”; I

8  J = 1

10 INPUT “FLOW:”; F, “FREQ:”; K

12 FOR L=1 TO K: N = N + F/(1 + I/100)^J: J = J+1

14 NEXT L

16 INPUT “MORE=1: “; L   // enter 1 to enter more cash flows, anything else to end entry

18 PRINT USING “#############.##”; “NPV: “; N

20 END

Example:

CF0:  -7,000,  Interest Rate:  8%

Flow 1:  2,000, Freq 1: 1  (enter 1 for MORE)

Flow 2:  1,500, Freq 2: 2  (enter 1 for MORE)

Flow 3:  2,500, Freq 3: 2  (we are at the end, enter anything other than 1 at MORE)

Result:

NPV:  2443.06

Sharp EL-5500 III Program: Synthetic Division

2 PAUSE “Synthetic Division”: CLEAR

4 PRINT “P(X)/(X – R)”: WAIT 59   // WAIT 59 is about 1 second

6 INPUT “DEGREE:”; N

8 DIM P(N): DIM Q(N)

10 FOR I=1 TO N

12 PRINT “COEF OF X^”; N-I

14 INPUT P(I): Q(I) = P(I)

16 NEXT I   // there is no line 18

20 INPUT “R:”; R

22 FOR I=0 TO N-1

24 Q(I+1) = R*Q(I) + P(I+1)

26 NEXT I

28 E = Q(N)

30 PRINT “Q(X) = “

32 FOR I=0 TO N-1

34 PRINT Q(I); “X^”; N-I-1

36 WAIT 150  // about 2.5 seconds

38 NEXT I

40 PRINT “+”; E; “/(X-“; R; “)”: END

Example:  (x^4 – 2*x^3 + 1) / (x – 1)

Degree: 4

Coefficients:  1, -2, 0, 0, 1

R: 1

Result: 1, -1, -1, -1, no remainder

x^3 – x^2 – x – x

Sharp EL-5500 III Program: Vector Basics

Cross product, dot product, norm of two vectors, angle between two vectors

2 PAUSE “Vector Basics”

4 INPUT “X1:”; X1, “Y1:”; Y1, “Z1:”; Z1  \\ vector 1

6 INPUT “X2:’; X2, “Y2:”; Y2, “Z2:”, Z2  \\ vector 2

8 C1 = Y1*Z2 – Y2*Z1: C2 = -X1*Z2 + X2*Z1: C3 = X1*Y2 – X2*Y1  // cross product

10 D = X1*X2 + Y1*Y2 + Z1*Z2  \\ dot product

12 N1 = √(SQU X1 + SQU Y1 + SQU Z1): N2 = √(SQU X2 + SQU Y2 + SQU Z2)  // norm, SQU is the x^2 key

14 DEGREE

16 A = ACS( D/(N1*N2))  // angle between vectors, ACS is ACOS

18 PRINT “CROSS X: ”; C1

20 PRINT “CROSS Y: ”; C2

22 PRINT “CROSS Z: ”; C3

24 PRINT “DOT: ”; D

26 PRINT “NORM V1: “; N1

28 PRINT “NORM V2: “; N2

30 PRINT “ANGLE: “; A

32 END

Example:  V1 = [-2, 3, 0] and V2 = [ 4, 2, -11]

Cross: [-33, -22, -16]

Dot: -2

Norm V1: 3.605551275

Norm V2: 11.87434209

Angle: 92.67749998°

Sharp EL-5500 III: Atwood Machine

M1:  mass hanging on the left side of the machine

M2:  mass hanging on the right side of the machine

The program asks to choose a unit system.  Enter 1 for US units (feet, pounds, seconds, g = 32.174 ft/s^2), anything else for SI units (meters, kilograms, seconds, g = 9.80665 m/s^2).

2 PAUSE “Atwood Machine”

4 INPUT “1: US, ELSE: SI “; I

6 IF I=1 THEN LET G=32.174

8 IF I<>1 THEN LET G=9.80665

10 INPUT “M1: “; M1, “M2: “; M2

12 A = (M1 – M2)*G / (M1 + M2)

14 T = M1 * (G-A)

16 PRINT “Accel.: “; A

18 PRINT “Tension: “: T

20 END

Example:  M1 = 11.82 kg, M2 = 9.38 kg, use SI units (enter anything other than 1 at the 1:US, ELSE:SI prompt)

Results:

Accel.: 1.128689906 m/s^2

Tension: 102.5734883 N

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.