113 lines
4.7 KiB
Markdown
113 lines
4.7 KiB
Markdown
---
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title: Cartesian Coordinate System
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TARGET DECK: Obsidian::STEM
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FILE TAGS: geometry::coordinates
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tags:
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- geometry
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---
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## Overview
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In plane analytic geometry, the **Cartesian coordinate system** uniquely specifies a point by a pair of real numbers called its **coordinates**. These coordinates represent signed distances to the point from two fixed perpendicular oriented lines called the **axes**. The point where the axes meet is called the **origin** and have coordinates $\langle 0, 0 \rangle$.
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%%ANKI
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Cloze
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The {$x$-coordinate} of a point is sometimes called its {abscissa}.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731184865785-->
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END%%
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%%ANKI
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Cloze
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The {$y$-coordinate} of a point is sometimes called its {ordinate}.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731184865791-->
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END%%
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%%ANKI
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Cloze
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The {origin} of a Cartesian coordinate system has coordinates $\langle 0, 0 \rangle$.
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Reference: “Cartesian Coordinate System,” in _Wikipedia_, October 21, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system](https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system&oldid=1252434514).
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<!--ID: 1731184876657-->
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END%%
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%%ANKI
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Basic
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Consider point $\langle x, y \rangle$. When does this point lie in the first quadrant?
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Back: When $x > 0$ and $y > 0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731184865794-->
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END%%
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%%ANKI
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Basic
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Consider point $\langle x, y \rangle$. When does this point lie in the second quadrant?
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Back: When $x < 0$ and $y > 0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731184865796-->
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END%%
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%%ANKI
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Basic
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Consider point $\langle x, y \rangle$. When does this point lie in the fourth quadrant?
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Back: When $x > 0$ and $y < 0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731184865799-->
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END%%
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%%ANKI
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Basic
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Consider point $\langle x, y \rangle$. When does this point lie in the third quadrant?
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Back: When $x < 0$ and $y < 0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731184865801-->
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END%%
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%%ANKI
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Basic
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The "vertical line test" of a Cartesian coordinate system is used to determine what?
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Back: Whether the tested graph depicts a function or not.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731185808236-->
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END%%
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%%ANKI
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Basic
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In Cartesian coordinate systems, why does the vertical line test work?
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Back: A function is single-valued. A vertical line that intersects a graph multiple times immediately contradicts this.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731185808243-->
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END%%
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## Cartesian Equations
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An equation that completely characters a figure within the Cartesian coordinate system is called a **Cartesian equation**.
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%%ANKI
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Basic
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What is a Cartesian equation?
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Back: An equation that completely characterizes a figure within the Cartesian coordinate system.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731185808246-->
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END%%
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%%ANKI
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Basic
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What is the Cartesian equation of a circle centered around the origin with radius $r$?
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Back: $x^2 + y^2 = r^2$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731185808249-->
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END%%
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%%ANKI
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Basic
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What figure does the following Cartesian equation characterize? $x^2 + y^2 = r^2$
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Back: A circle with radius $r$ centered around the origin.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1731185808252-->
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END%%
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## Bibliography
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* “Cartesian Coordinate System,” in _Wikipedia_, October 21, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system](https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system&oldid=1252434514).
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* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). |