Mathematical Visualisation using GeoGebra

Juan Carlos Ponce Campuzano

Linear regression: Residuals

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GTXF J8AA

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1. Exploring systems of linear equations in 3D

Consider the system corresponding to three planes

\begin{array}{rcl} x+ 5y &=& 1+ 2z\\ x+z &=& 3y+3\\ 8y- \lambda &=& 3z \end{array}

\(\lambda\) is a parameter.

  1. Use Gaussian elimination to determine the value of \(\lambda\) for which this system has infinitely many solution.
  2. Use part a) to determine the infinitely many solutions. Express your answer in the form of a vector equation

1. Exploring systems of linear equations in 3D

\begin{array}{rcl} x+ 5y &=& 1+ 2z\\ x+z &=& 3y+3\\ 8y- \lambda &=& 3z \end{array}
\begin{array}{rrrrrrr} x&+& 5y &- &2z &=& 1\\ x&-&3y &+&z &=& 3\\ 0x &+& 8y&-& 3z &=& \lambda \end{array}
\left( \begin{array}{rrr|r} 1 & 5 & -2 & 1\\ 1 & -3 & 1 & 3\\ 0 & 8 & -3 & \lambda \end{array} \right)
\left( \begin{array}{rrr|r} 1 & 5 & -2 & 1\\ 0 & -8 & 3 & 2\\ 0 & 8 & -3 & \lambda \end{array} \right)
R_2-R_1
\lambda = -2
R_1\\ R_2

the system is consistent with \(\infty\) solutions

1. Exploring systems of linear equations in 3D

lambda = Slider(-10, 10, 0.1)
eq1: x + 5y = 1 + 2z
eq2: x + z= 3y + 3
eq3: 8y - lambda = 3z

Solve({eq1, eq2, eq3})

line: X = (9/4, -1/4, 0) + t * (1/8, 3/8, 1)

🔗 Solve                      

🔗 Slider                      

🔗 Equations of lines

2. Differential equations and slope fields

The slope fied for the differential equation \(\dfrac{dy}{dx}=\dfrac{-0.5(y-4)}{x},\) with \(x\neq 0,\) where \(-6\leq x\leq 6\) and \(-6\leq y \leq 6\) is shown below.

  1. Determine the value of the slope at point \(A\).
  2. Use the slope field to sketch the solution curve given \(x=-6,\) \(y=3.5\)

2. Differential equations and slope fields

n = Slider(10, 30, 1)
s = Slider(0.1, 1, 0.1)

F(x, y) = -0.5 * (y - 4)/x

SlopeField(F, n, s, -6, -6, 6, 6)

A = (0, 0)

SolveODE(F, x(A), y(A), 6, 0.01)

SolveODE(F, x(A), y(A), -6, 0.01)

🔗 SlopeField

🔗 SolveODE 

3. Roots of complex numbers

Represent in the Argand diagram the solutions of

z^{1/n}
\text{with}\;k=0,1,\ldots, n-1
z^4= 16 \,\text{cis}\left(\dfrac{2\pi}{3}\right)
= 16 \cos\left(\dfrac{2\pi}{3}\right) + i\,16\sin\left(\dfrac{2\pi}{3}\right)
=\sqrt[n]{r}\left[\cos\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right) \right]

3. Roots of complex numbers

Represent in the Argand diagram the solutions of

z^{1/n}
\text{with}\;k=0,1,\ldots, n-1
z^4= 16 \,\text{cis}\left(\dfrac{2\pi}{3}\right)
= 16 \cos\left(\dfrac{2\pi}{3}\right) + i\,16\sin\left(\dfrac{2\pi}{3}\right)
= r^{1/n} \exp\Bigg(i \left( \frac{\theta}{n}+\frac{2\pi k}{n} \right)\Bigg)

3. Roots of complex numbers

z_1 = i
u = Vector(z_1)
r = abs(z_1)
theta = arg(z_1)
n = Slider(1, 8, 1)

L = Sequence(r^(1/n)*exp(i * ( theta/n + 2pi * k / n )), k, 0, n-1)

Sequence(Vector(Element(L, k)), k, 1, Length(L))

# Alternatively
Zip(Vector(Point), Point, L)

🔗 Sequence      

🔗 Vector            

🔗 Zip (advance)

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