Orienting Yourself: The Use of Coordinates

Introduction to Coordinate Systems

• Definition: A system of coordinates is a structured framework, similar to grid lines on a map or graph paper, that enables the use of numbers to describe the exact physical locations of points or objects.

• Historical Roots in Bharat:     * Sindhu-Sarasvati Civilisation: Historically, grid-based thinking and geometry for defining points in space were used thousands of years ago in large-scale urban planning. City streets were constructed with precision in North-South and East-West directions at uniform distances of approximately $10\,m$ apart. This allowed individuals, such as merchants, to locate shops or warehouses by counting North-South and East-West units from the city center.     * Baudhāyana (c. 800 C.E.): Utilized East-West and North-South lines for deep geometric constructions. He developed the Baudhāyana-Pythagoras Theorem, which laid the foundation for coordinate geometry.

• Historical Development for Navigation:     * Ujjayinī: Historically described in ancient works (early Sidhāntas as early as 4th century BCE) as the point marking the central longitude meridian from which all other locations were measured.     * Ptolemy (c. 150 BCE): A Greek mathematician who built on the work of Hipparchus to describe latitudes and longitudes for thousands of locations, including Ujjayinī, which he referred to as 'Ozine'.     * Aryabhata (c. 499 CE): Replaced Greek 'chords' with 'sines', facilitating the calculation of coordinates for cities or stars. He mapped the sky using Celestial Coordinates, measuring distances from the ecliptic (the sun's path).     * Brahmagupta (c. 628 CE): Formalized the concept of zero and negative numbers as algebraic entities. In modern coordinate systems, the 'origin' is zero and 'negative axes' represent values less than zero; the four-quadrant Cartesian plane relies on this work.

• Global Knowledge Transfer:     * Arabic Influence: Brahmagupta's work was translated into Arabic as the Sindhind. The Ujjayinī meridian entered Arabic geography as 'Arin,' serving as the zero-longitude reference.     * Al-Biruni (c. 1000 CE): Travelled to India, studied the Siddhāntas, and used Indian trigonometric methods to calculate the coordinates of various Asian cities. He also perfected the 'astrolabe,' a device for sailors to locate coordinates via stars.     * Omar Khayyam (c. 1100 CE): An expert in the Indian decimal system and algebraic formalism, he was the first to solve algebraic problems using geometry by interpreting them as coordinates on a plane.     * European Formalization: These concepts reached Europe in the 12th century. Following Fermat (1636 CE), René Descartes (1637 CE) formalized that any point in a two-dimensional plane could be defined by two numbers representing distances from two perpendicular axes. This linked geometry and algebra through equations and shapes.

Practical Application: The Story of Reiaan and Shalini

• Scenario: Reiaan is settling into a new home in a new city. Because Reiaan cannot see, his sister Shalini uses Coordinate Geometry to help him navigate their room.

• Visual Aid/Map: Shalini created a rectangular grid using pins and threads with a scale of $1\,cm : 1\,\text{foot}$.     * Room Layout (Fig 1.1):         * Bedroom: $12\,ft \times 10\,ft$.         * Bathroom: $6\,ft \times 9\,ft$.         * Wardrobe: $4\,ft \times 2\,ft$.

• Tactile Navigation: Points for corners were marked with pins, and thick wool connected them so Reiaan could feel the positions with his fingers.

• Constraint: The map only shows the floor layout; therefore, the position of windows cannot be marked on this specific map.

The 2-D Cartesian Coordinate System

• Structure: The system uses two lines at right angles to each other to mark points in two-dimensional space ($2\text{-D}$ space).

• x-axis: The horizontal line.

• y-axis: The vertical line.

• Origin (O): The point of intersection where the x-axis and y-axis meet. Its coordinates are $(0, 0)$.

• Coordinate Axes: The plural of axis, used to locate points using 'coordinates'.

• Units and Directionality:     * Distances are marked in equal units on both axes.     * Distances to the right of $O$ (along the x-axis) or upwards from $O$ (along the y-axis) are positive.     * Distances to the left of $O$ or downwards from $O$ are negative.

• Notation: A point is represented as $(x, y)$.     * $x$: The perpendicular distance of point $P$ from the y-axis, measured along the x-axis (x-coordinate).     * $y$: The perpendicular distance of point $P$ from the x-axis, measured along the y-axis (y-coordinate).     * Example: Point $B = (4.5, 0)$ lies on the x-axis, $4.5$ units to the right of $O$. Point $G = (0, -4.5)$ lies on the y-axis, $4.5$ units downward from $O$.

The Cartesian Plane and Quadrants

• The Plane: Also known as the coordinate plane or the $xy$-plane.

• Quadrants: The axes divide the plane into four parts, numbered I through IV:     * Quadrant I: Both x- and y-coordinates are positive $(+, +)$.     * Quadrant II: x-coordinate is negative, y-coordinate is positive $(-, +)$.     * Quadrant III: Both x- and y-coordinates are negative $(-, -)$.     * Quadrant IV: x-coordinate is positive, y-coordinate is negative $(+, -)$.

• Example Points:     * Point $S(3, -5)$ is in Quadrant IV.     * Point $Q(-5, 3)$ is in Quadrant II.

Distance Between Two Points in the 2-D Plane

• Horizontal and Vertical Distances: Distance on axes or parallel to axes is found by the absolute difference between coordinates.     * Distance between $(x_1, y)$ and $(x_2, y)$ is $|x_2 - x_1|$.     * Distance between $(x, y_1)$ and $(x, y_2)$ is $|y_2 - y_1|$.

• General Distance Formula: Derived from the Baudhāyana-Pythagoras Theorem.     * Given points $A(x_1, y_1)$ and $D(x_2, y_2)$.     * Horizontal distance ($x\text{-shift}$): $CD = x_2 - x_1$.     * Vertical distance ($y\text{-shift}$): $AC = y_2 - y_1$.     * The distance $AD$ is the hypotenuse: $AD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

• Worked Example (Triangle ADM):     * $A(3, 4)$, $D(7, 1)$, and $M(9, 6)$.     * Calculating $AD$:         * $x\text{-shift} = 7 - 3 = 4$.         * $y\text{-shift} = 4 - 1 = 3$.         * $AD = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\,\text{units}$.     * Calculating $DM$:         * $DM = \sqrt{(9 - 7)^2 + (6 - 1)^2} = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}\,\text{units}$.     * Calculating $MA$:         * $MA = \sqrt{(9 - 3)^2 + (6 - 4)^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40}\,\text{units}$.

Reflection in the Coordinate Plane

• Concept: Reflecting a shape across an axis changes specific coordinates while preserving side lengths (congruence).

• Reflection across the y-axis:     * Original triangle $ADM$ with $A(3, 4)$, $D(7, 1)$, and $M(9, 6)$.     * Image triangle $A'D'M'$ coordinates:         * $A'(-3, 4)$.         * $D'(-7, 1)$.         * $M'(-9, 6)$.     * Verification of Length: $C'D' = -3 - (-7) = 4$. $A'C' = 4 - 1 = 3$. $A'D' = \sqrt{4^2 + 3^2} = 5\,\text{units}$. This confirms that reflection preserves lengths.

Think and Reflect Key Questions

1. What is the x-coordinate of a point on the y-axis? The x-coordinate is always $0$.

2. Point On x-axis: The y-coordinate is always $0$.

3. Does $Q(y, x)$ ever coincide with $P(x, y)$? Only if $x = y$.

4. Equality Requirement: $(x, y) = (y, x)$ if and only if $x = y$. If $x \neq y$, then $(x, y) \neq (y, x)$.

Detailed Exercise & Case Study Data

• Room Map Exercise 1.2 Data:     * Corners: $O(0, 0)$, $A(12, 0)$, $B(12, 10)$, $C(0, 10)$.     * Door $D_1 R_1$: $R_1$ is at $(11.5, 0)$. If the door moves from $A(12, 0)$ towards $11.5$, its width is $0.5\,units$ ($0.5\,ft$).     * Bathroom door: $B_1(0, 1.5)$ and $B_2(0, 4)$. Width = $4 - 1.5 = 2.5\,units$. This is significantly wider than the room door described.

• Study Table Problem: Three feet at $(8, 9)$, $(11, 9)$, and $(11, 7)$. The fourth foot for a rectangular table would be at $(8, 7)$. The table width is $11 - 8 = 3\,ft$ and length/depth is $9 - 7 = 2\,ft$.

• Dining Room: Length $18\,ft$, width $15\,ft$, extending from $P$ to $A$.

• Computer Graphics Screen: $800 \times 600\,\text{pixels}$. Origin at bottom-left corner.     * Icon A: Centre $(100, 150)$, radius $80\,pixels$.     * Icon B: Centre $(250, 230)$, radius $100\,pixels$.

• City Model: Two main roads intersect at the center (North-South and East-West). Streets are $200\,m$ apart. Scale used: $1\,cm = 200\,m$. Intersection $(2, 5)$ refers to the intersection of 2nd street N-S and 5th street E-W.

• Trisection Example: Find coordinates of $P$ and $Q$ that trisect segment $AB$ where $A(4, 7)$ and $B(16, -2)$.

• Circle Center Example: Origin $O(0, 0)$. Checking points $A(1, -8)$, $B(-4, 7)$, and $C(-7, -4)$. Radius is determined by the distance formula from the origin to any point on the circle.

Questions & Discussion

• Q: What would a coordinate system be like without negative numbers?

• A: It would be limited to Quadrant I (positive x and y). Such a system could not locate all points in a full $2\text{-D}$ plane, specifically those to the left of or below the designated origin.

• Q: Are points $M(-3, -4)$, $A(0, 0)$, and $G(6, 8)$ on the same straight line?

• A: This can be checked by verifying if the slope between $MA$ and $AG$ is identical or by using the distance formula to see if $MA + AG = MG$.