Coordinate System Board Game: Interactive Demo

Dual 20-Face Dice Roller, Pythagorean Origin Distance & Vector Triangulation

Current Point (X, Y)
Vector Distance (d)
Previous Plotted Marks

Dice Station & Point Calculations

X-Axis (Green Die)
0
Y-Axis (White Die)
0
Plotted Coordinate: (0, 0)
Quadrant / Location: Origin
Complex Form ($z = x + yi$): 0 + 0i
Pythagorean Formula: $d = \sqrt{x^2 + y^2}$
Exact Distance: √0 = 0.00

Roll History

# X (Green) Y (White) Quadrant Distance ($d$)

Mathematical Foundations: Vectors & Complex Numbers

1. Definition of Vectors

A vector is a mathematical quantity possessing both magnitude (length) and direction. In the Cartesian coordinate plane:

  • Component Form: $\vec{v} = \langle x, y \rangle = x\hat{i} + y\hat{j}$, where $\hat{i}$ and $\hat{j}$ represent unit vectors along the $X$ and $Y$ axes.
  • Vector Magnitude (Euclidean Norm): The physical straight-line distance from the origin $(0,0)$ to $(x, y)$, determined by the Pythagorean theorem: $$||\vec{v}|| = \sqrt{x^2 + y^2}$$
  • Direction Angle ($\theta$): Measured counterclockwise from the positive $X$-axis: $\theta = \arctan\left(\frac{y}{x}\right)$.

2. Definition of Complex Numbers

A complex number is a number that can be expressed in the form $z = a + bi$, where:

  • Real Part ($\text{Re}(z)$): $a \in \mathbb{R}$, mapped along the horizontal real axis ($X$).
  • Imaginary Part ($\text{Im}(z)$): $b \in \mathbb{R}$, mapped along the vertical imaginary axis ($Y$).
  • Imaginary Unit: Defined by $i = \sqrt{-1}$, where $i^2 = -1$.
  • Modulus (Absolute Value): The distance from the origin to $z$ on the complex plane: $$|z| = \sqrt{a^2 + b^2}$$
  • Complex Conjugate: Denoted $\bar{z} = a - bi$, reflecting $z$ across the real axis.

3. Algebraic Operations for Complex Numbers

Let $z_1 = a + bi$ and $z_2 = c + di$ be two complex numbers:

Addition ($z_1 + z_2$)

Add the real components and imaginary components independently:

$$(a + bi) + (c + di) = (a + c) + (b + d)i$$

Subtraction ($z_1 - z_2$)

Subtract the real components and imaginary components independently:

$$(a + bi) - (c + di) = (a - c) + (b - d)i$$

Multiplication ($z_1 \cdot z_2$)

Expand using the distributive FOIL method, substituting $i^2 = -1$:

$$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$

Division ($\frac{z_1}{z_2}$)

Multiply numerator and denominator by the conjugate of the denominator ($\bar{z}_2 = c - di$):

$$\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$