%%{init: {"quadrantChart": {"xAxisPosition": "bottom", "yAxisPosition": "left", "xRange": [-10, 10], "yRange": [-10, 10]}}} "titlePadding": 20}}}%%

quadrantChart
    title Cartesian Plane
    x-axis x
    y-axis y

    P1: [0.75, 0.75]
    P2: [0.45, 0.23]
    Origin: [0, 0] radius: 5, color: #444444, stroke-color: #ffffff, stroke-width: 3px
    x : [0.1, 0] radius: 5, color: #666666, stroke-color: #ff0000, stroke-width: 3px
    y : [0, 0.1] radius: 5, color: #666666, stroke-color: #00ff00, stroke-width: 3px

The great Descartes made a connection between algebra and geometry. He developed the Cartesian Plane.

In it, we can define algebraically different geometric points.

Here are two simple points:

• $P_1 = (x_1, y_1)$
• $P_2 = (x_2, y_2)$

We can then define the line that goes through both points.

Where:

• $m = \frac{y_2-y_1}{x_2-x_1}$ is the slope of the line.
• $m = \tan{𝜃}$, where $𝜃$ is the angle with $𝑥̂$.

The equation of the line through the point $P = (x_1, y_1, z_1)$ and pointing in the direction of the vector $\vec{d} = a\hat{i} + b\hat{j} + c\hat{k}$ is:

Where:

• $R = (x, y, z)$ is the general point on the line.
• $t$ is a parameter that takes on all real values.

In coordinate form, the equations are:

The parametric equations of the line l through the points $P = (x_1, y_1, z_1)$ and $Q = (x_2, y_2, z_2)$ are

%%{init: {"quadrantChart": {"pointRadius": 3, "pointTextPadding": 13, "titlePadding": 20}}}%%

quadrantChart
    title Line
    x-axis x
    y-axis y
    P1: [0.75, 0.6] radius: 5, color: #7560ff
    P2: [0.45, 0.23] radius: 5, color: #4523ff
    D: [0.3, 0.37] radius: 5, color: #a57c00

    l1: [0.48, 0.267] color: #00FFFF

    l2: [0.51, 0.304] color: #00FFFF

    l3: [0.54, 0.34099999999999997] color: #00FFFF

    l4: [0.5700000000000001, 0.378] color: #00FFFF

    l5: [0.6000000000000001, 0.41500000000000004] color: #00FFFF

    l6: [0.63, 0.45199999999999996] color: #00FFFF

    l7: [0.66, 0.489] color: #00FFFF

    l8: [0.69, 0.526] color: #00FFFF

    l9: [0.72, 0.5630000000000001] color: #00FFFF

# for points 
(x1, y1) = (0.75, 0.6) # P1
(x2, y2) = (0.45, 0.23) # P2
L = 10 # Line Points
dL = 1/L # line segment size
(dx, dy) = (x1-x2, y1-y2) # Vector distance

for t in range(0,L+1):
    print(f"    l{t}: [{x2 + (dx)*t*dL}, {y2+(dy)*t*dL}] color: #00FFFF\n")

If an object has a (constant) velocity vector $\vec{v}$, then in $t$ units of time the resulting displacement vector of the object is

Consider a seagull that flies in calm air with velocity vector $\vec{v}$. If a wind comes up with velocity $\vec{w}$ and the seagull continues flying the same way, its actual velocity will be $\vec{v}$ + $\vec{w}$. One can see the direction of the vector $\vec{v}$ because it points along the axis of the seagull. By comparing the direction of actual motion with the direction of $\vec{v}$, you can get an idea of the wind direction

Ordered List of elements that are part of a Vector Space.

Defined from an origin

$𝐯̂$ :

• $|𝐯̂|=1$
• $𝐯̂∥𝐯⃗$

$𝛂𝐯⃗$ :

• $| 𝛂𝐯⃗ |= 𝛂| 𝐯⃗ |$
• $𝛂𝐯⃗ ∥𝐯⃗$

𝐮⃗+𝐯⃗ = 𝐰⃗

• 𝐰ᵢ = 𝐮ᵢ+𝐯ᵢ

𝐮⃗,𝐯⃗,𝐰⃗∈ 𝕌ⁿ

𝐮⃗+𝐯⃗= 𝐯⃗+𝐮⃗

𝐮⃗+𝐯⃗+𝐰⃗ = 𝐮⃗+ (𝐯⃗+𝐰⃗) = (𝐮⃗+𝐯⃗)+𝐰⃗

𝐮⃗+ 𝟎⃗ = 𝐮⃗

𝐮⃗- 𝐮⃗ = 𝟎⃗

|𝐯⃗∣≥0

|𝐯⃗|=0⟷𝐯⃗= 𝟎⃗

|𝐯⃗∣= |-𝐯⃗∣

|𝐮⃗+𝐯⃗|≤ |𝐮⃗∣+|𝐯⃗∣

𝛼𝐯⃗ = 𝐯⃗𝛼

𝛼(𝛽𝐯⃗)= (𝛼𝛽)𝐯⃗

𝛼(𝐮⃗+𝐯⃗)= 𝛼𝐮⃗+𝛼𝐯⃗

(𝛼+𝛽)𝐯⃗ = 𝛼𝐯⃗+ 𝛽𝐯⃗

1𝐯⃗= 𝐯⃗

0𝐯⃗= 𝟎⃗ 𝛼𝟎⃗= 𝟎⃗

𝐯⃗∕ 𝛼 = 𝐯⃗(1 ∕ 𝛼)

𝐯̂=𝐯⃗∕ |𝐯⃗|

𝕌ⁿ

• 𝟎⃗∈ 𝕌ⁿ
• 𝐯⃗ ∈ 𝕌ⁿ⟶ 𝛼𝐯⃗ ∈ 𝕌ⁿ
• 𝐮⃗,𝐯⃗∈ 𝕌ⁿ ⟶ [𝐮⃗+𝐯⃗]∈ 𝕌ⁿ

• Linear: [𝛼𝐮⃗+𝛽𝐯⃗]∈ 𝕌ⁿ
  • 𝛼,𝛽∈𝔸

𝕊ⁿ ⊂ 𝕌ⁿ

Subspace 𝕊ⁿ: Space conditions

Equation of a Sphere.

• Centre: $𝖢(h,k,l)$
• Point: $𝖯(x,y,z)$

Given a Point $\color{BlueGreen} 𝖯(x_0,y_0,z_0)$ and a vector $\color{CornflowerBlue}𝐯⃗ = ⟨v_x, v_y, v_z⟩$ a line can be described by the equations:

Or in general

• $A = 0$ OR $C=0$
  • Parabola
• $A = C$
  • Circunference
• $sign(A) = sign(C)$
  • Elipse
• $sign(A) = -sign(C)$
  • Hiperbola

• $B^2 - 4AC = 0$
  • Parabola
• $B^2 - 4AC < 0$
  • Elipse
• $B^2 - 4AC > 0$
  • Hiperbola

With $\color{Cerulean}𝖥$ the fixed point, Focus, $\color{NavyBlue}𝓁$ the straight line, Directrix, and $\color{RedOrange}ℯ$ a factor called Excentricity, then the set of all points $\color{ProcessBlue}𝑃$ such as

With:

• ${\color{RedOrange}ℯ} = 1$ for Parábolas
• ${\color{RedOrange}ℯ} < 1$ for Elipses
• ${\color{RedOrange}ℯ} > 1$ for Hiperbola

Given a Point $\color{BlueGreen} 𝖯_0(x_0,y_0,z_0)$ of the plane and the Normal Vector to the surface $\color{LightBlue}𝐧⃗ = ⟨n_x, n_y, n_z⟩$ the Plane is defined by

If we call the Position Point $\color{ProcessBlue}𝖷(x,y,z)$ :

Having three points

we can calculate the Normal Vector to the plane

The equation of the plane through $(x_0, y_0, z_0)$ that has a normal vector

$\vec{n} = A\hat{x} + B\hat{y} + C \hat{z} .$ is:

transformable to the system:

That we can generalize as

%%{init: {"quadrantChart": {"xAxisPosition": "bottom", "yAxisPosition": "left", "xRange": [-10, 10], "yRange": [-10, 10]}}} "titlePadding": 20}}}%%

quadrantChart
    title Point(7,6) and N(4, 2.3)
    x-axis x
    y-axis y
    
    X: [0.7, 0.6]
    N: [0.4, 0.23]

    S7_6: [0.0, 0.0] color: #0706FF

    S7_7: [0.0, 0.22999999999999998] color: #0707FF

    S7_8: [0.0, 0.45999999999999996] color: #0708FF

    S7_9: [0.0, 0.69] color: #0709FF

    S7_10: [0.0, 0.9199999999999999] color: #0710FF

    S8_6: [0.4, 0.0] color: #0806FF

    S8_7: [0.4, 0.22999999999999998] color: #0807FF

    S8_8: [0.4, 0.45999999999999996] color: #0808FF

    S8_9: [0.4, 0.69] color: #0809FF

    S8_10: [0.4, 0.9199999999999999] color: #0810FF

    S9_6: [0.8, 0.0] color: #0906FF

    S9_7: [0.8, 0.22999999999999998] color: #0907FF

    S9_8: [0.8, 0.45999999999999996] color: #0908FF

    S9_9: [0.8, 0.69] color: #0909FF

    S9_10: [0.8, 0.9199999999999999] color: #0910FF

scale = 10
(x0, y0) = (7, 6) # Point in Surface
(a, b) = (4, 2.3) # Normal
S = 10 # Origin Line Points
dS = 1/S # line segment size

for x in range(0,S+1):
    for y in range(0,S+1):
        X = a*(x - x0 )/scale
        Y = b*(y - y0 )/scale
        if X<= 1 and X>=0 and Y<= 1 and Y>=0:
            print(f"    S{x}_{y}: [{X }, {Y }] color: #{x:02}{y:02}FF\n")