Give The Coordinates Of The Following Points

Here's a comprehensive guide on understanding and determining the coordinates of points in various coordinate systems. This exploration covers Cartesian coordinates, polar coordinates, spherical coordinates, and cylindrical coordinates, providing you with the knowledge to confidently locate points in both two and three dimensions.

Understanding Coordinate Systems

A coordinate system is a method of uniquely identifying the position of a point in space. It provides a framework using numbers, called coordinates, that represent the point's location relative to a fixed reference point, often called the origin. Different coordinate systems are better suited for different applications, depending on the geometry of the problem.

Cartesian Coordinates (Rectangular Coordinates)

The Cartesian coordinate system, also known as the rectangular coordinate system, is the most commonly used system. It employs two or three mutually perpendicular axes to define a point's position.

• Two Dimensions (2D): In a 2D Cartesian plane, we have two axes: the x-axis (horizontal) and the y-axis (vertical). A point is represented by an ordered pair (x, y), where x is the horizontal distance from the origin along the x-axis, and y is the vertical distance from the origin along the y-axis.

• Three Dimensions (3D): In a 3D Cartesian space, we add a third axis, the z-axis, which is perpendicular to both the x-axis and the y-axis. A point is represented by an ordered triple (x, y, z), where x, y, and z are the distances from the origin along the respective axes.

Polar Coordinates

Polar coordinates are used to represent a point in a two-dimensional plane using a distance from the origin (pole) and an angle from a reference direction (usually the positive x-axis).

• A point in polar coordinates is represented as (r, θ), where r is the radial distance from the origin to the point, and θ is the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point.
• r is always non-negative.
• θ is typically expressed in radians or degrees.

Cylindrical Coordinates

Cylindrical coordinates extend the concept of polar coordinates to three dimensions. They are particularly useful when dealing with objects that have cylindrical symmetry.

• A point in cylindrical coordinates is represented as (r, θ, z), where (r, θ) are the polar coordinates of the point's projection onto the xy-plane, and z is the height of the point above the xy-plane.
• Essentially, cylindrical coordinates are polar coordinates with an added z-coordinate indicating height.

Spherical Coordinates

Spherical coordinates provide a way to represent a point in three dimensions using a radial distance from the origin, an azimuthal angle, and a polar angle. They are particularly useful when dealing with objects that have spherical symmetry.

• A point in spherical coordinates is represented as (ρ, θ, φ), where:
  • ρ (rho) is the radial distance from the origin to the point (always non-negative).
  • θ (theta) is the azimuthal angle, measured counterclockwise from the positive x-axis to the projection of the point onto the xy-plane (same as the angle in polar and cylindrical coordinates).
  • φ (phi) is the polar angle, measured from the positive z-axis to the point (ranges from 0 to π radians or 0 to 180 degrees).

Determining Coordinates: A Step-by-Step Guide

The process of finding the coordinates of a point depends on the coordinate system being used and the information available about the point's location. Here's a breakdown of how to determine coordinates in each system.

1. Cartesian Coordinates

Given a point visually on a graph:

1. Identify the Axes: Determine the x-axis and y-axis (in 2D) or the x, y, and z-axes (in 3D).
2. Find the x-coordinate: Draw a vertical line from the point to the x-axis. The value where the line intersects the x-axis is the x-coordinate.
3. Find the y-coordinate: Draw a horizontal line from the point to the y-axis. The value where the line intersects the y-axis is the y-coordinate.
4. Find the z-coordinate (if in 3D): Imagine or draw a line from the point perpendicular to the xy-plane and intersecting the z-axis. The value where the line intersects the z-axis is the z-coordinate.
5. Write the Coordinates: Express the coordinates as an ordered pair (x, y) in 2D or an ordered triple (x, y, z) in 3D.

Example (2D): A point is located 3 units to the right of the origin and 2 units above the origin. Its Cartesian coordinates are (3, 2).

Example (3D): A point is located 1 unit to the left of the origin along the x-axis, 4 units in front of the origin along the y-axis, and 5 units above the origin along the z-axis. Its Cartesian coordinates are (-1, 4, 5).

2. Polar Coordinates

Given a point visually on a polar grid:

1. Identify the Pole and Polar Axis: The pole is the origin, and the polar axis is usually the positive x-axis.
2. Measure the Radial Distance (r): Measure the distance from the pole to the point. This is the value of r.
3. Measure the Angle (θ): Measure the angle counterclockwise from the polar axis to the line segment connecting the pole to the point. This is the value of θ.
4. Write the Coordinates: Express the coordinates as (r, θ).

Converting from Cartesian to Polar Coordinates:

If you have the Cartesian coordinates (x, y), you can convert them to polar coordinates (r, θ) using the following formulas:

• r = √(x² + y²)
• θ = arctan(y/x) (Be careful with the quadrant of the point when using the arctangent function. Consider the signs of x and y to determine the correct quadrant.)

Converting from Polar to Cartesian Coordinates:

If you have the polar coordinates (r, θ), you can convert them to Cartesian coordinates (x, y) using the following formulas:

• x = r cos(θ)
• y = r sin(θ)

Example:

• Cartesian to Polar: The point (3, 4) in Cartesian coordinates can be converted to polar coordinates as follows:
  • r = √(3² + 4²) = √(9 + 16) = √25 = 5
  • θ = arctan(4/3) ≈ 0.927 radians (or 53.13 degrees)
  • Therefore, the polar coordinates are approximately (5, 0.927) or (5, 53.13°).
• Polar to Cartesian: The point (6, π/3) in polar coordinates can be converted to Cartesian coordinates as follows:
  • x = 6 * cos(π/3) = 6 * (1/2) = 3
  • y = 6 * sin(π/3) = 6 * (√3/2) = 3√3
  • Therefore, the Cartesian coordinates are (3, 3√3).

3. Cylindrical Coordinates

Given a point visually in a cylindrical coordinate system:

1. Identify the r, θ, and z Axes: The z-axis is the same as in Cartesian coordinates. The r and θ are defined as in polar coordinates, considering the projection of the point onto the xy-plane.
2. Find r and θ: Project the point onto the xy-plane and determine the polar coordinates (r, θ) of that projection.
3. Find z: Determine the height of the point above the xy-plane. This is the z-coordinate.
4. Write the Coordinates: Express the coordinates as (r, θ, z).

Converting from Cartesian to Cylindrical Coordinates:

Given Cartesian coordinates (x, y, z), you can convert to cylindrical coordinates (r, θ, z) using:

• r = √(x² + y²)
• θ = arctan(y/x) (Remember to consider the quadrant!)
• z = z (The z-coordinate remains the same)

Converting from Cylindrical to Cartesian Coordinates:

Given cylindrical coordinates (r, θ, z), you can convert to Cartesian coordinates (x, y, z) using:

• x = r cos(θ)
• y = r sin(θ)
• z = z (The z-coordinate remains the same)

Example:

• Cartesian to Cylindrical: The point (1, 1, 2) in Cartesian coordinates converts to cylindrical coordinates as:
  • r = √(1² + 1²) = √2
  • θ = arctan(1/1) = π/4 (or 45°)
  • z = 2
  • The cylindrical coordinates are (√2, π/4, 2) or (√2, 45°, 2).
• Cylindrical to Cartesian: The point (2, π/6, 3) in cylindrical coordinates converts to Cartesian coordinates as:
  • x = 2 * cos(π/6) = 2 * (√3/2) = √3
  • y = 2 * sin(π/6) = 2 * (1/2) = 1
  • z = 3
  • The Cartesian coordinates are (√3, 1, 3).

4. Spherical Coordinates

Given a point visually in a spherical coordinate system:

1. Identify the ρ, θ, and φ Axes: ρ is the radial distance from the origin. θ is the same azimuthal angle as in cylindrical and polar coordinates. φ is the polar angle from the positive z-axis.
2. Find ρ: Measure the distance from the origin to the point.
3. Find θ: Project the point onto the xy-plane and determine the angle θ as in polar or cylindrical coordinates.
4. Find φ: Measure the angle from the positive z-axis to the line segment connecting the origin to the point.
5. Write the Coordinates: Express the coordinates as (ρ, θ, φ).

Converting from Cartesian to Spherical Coordinates:

Given Cartesian coordinates (x, y, z), you can convert to spherical coordinates (ρ, θ, φ) using:

• ρ = √(x² + y² + z²)
• θ = arctan(y/x) (Remember to consider the quadrant!)
• φ = arccos(z/ρ)

Converting from Spherical to Cartesian Coordinates:

Given spherical coordinates (ρ, θ, φ), you can convert to Cartesian coordinates (x, y, z) using:

• x = ρ sin(φ) cos(θ)
• y = ρ sin(φ) sin(θ)
• z = ρ cos(φ)

Converting from Cylindrical to Spherical Coordinates:

Given cylindrical coordinates (r, θ, z), you can convert to spherical coordinates (ρ, θ, φ) using:

• ρ = √(r² + z²)
• θ = θ (The azimuthal angle remains the same)
• φ = arccos(z/ρ) = arccos(z/√(r² + z²))

Converting from Spherical to Cylindrical Coordinates:

Given spherical coordinates (ρ, θ, φ), you can convert to cylindrical coordinates (r, θ, z) using:

• r = ρ sin(φ)
• θ = θ (The azimuthal angle remains the same)
• z = ρ cos(φ)

Example:

• Cartesian to Spherical: The point (1, 1, √2) in Cartesian coordinates converts to spherical coordinates as:
  • ρ = √(1² + 1² + (√2)²) = √(1 + 1 + 2) = √4 = 2
  • θ = arctan(1/1) = π/4 (or 45°)
  • φ = arccos(√2 / 2) = π/4 (or 45°)
  • The spherical coordinates are (2, π/4, π/4) or (2, 45°, 45°).
• Spherical to Cartesian: The point (4, π/3, π/6) in spherical coordinates converts to Cartesian coordinates as:
  • x = 4 * sin(π/6) * cos(π/3) = 4 * (1/2) * (1/2) = 1
  • y = 4 * sin(π/6) * sin(π/3) = 4 * (1/2) * (√3/2) = √3
  • z = 4 * cos(π/6) = 4 * (√3/2) = 2√3
  • The Cartesian coordinates are (1, √3, 2√3).

Practical Applications

Understanding coordinate systems is crucial in various fields:

• Navigation: GPS systems use coordinate systems (primarily latitude, longitude, and altitude, which are related to spherical coordinates) to pinpoint locations on Earth.
• Computer Graphics: Coordinate systems are fundamental in creating and manipulating 3D models and scenes.
• Physics: Many physical phenomena are described using coordinate systems, such as projectile motion, electromagnetism, and fluid dynamics.
• Engineering: Coordinate systems are used in surveying, construction, and design to accurately represent and analyze structures.
• Astronomy: Coordinate systems are essential for mapping the positions of stars, planets, and other celestial objects.

Common Mistakes to Avoid

• Quadrant Errors in Arctangent: Always check the signs of x and y when using the arctangent function to determine the correct quadrant for the angle θ.
• Units: Ensure that angles are consistently measured in either radians or degrees and that distances are measured in the appropriate units.
• Confusing φ and θ in Spherical Coordinates: Remember that φ is the angle from the positive z-axis, while θ is the azimuthal angle in the xy-plane.
• Forgetting the Order of Coordinates: The order of coordinates is crucial. (x, y) is different from (y, x), and (ρ, θ, φ) is different from (ρ, φ, θ).

Conclusion

Mastering coordinate systems is fundamental to understanding spatial relationships and representing points in space accurately. By understanding the principles behind Cartesian, polar, cylindrical, and spherical coordinates, you can confidently navigate a wide range of applications in mathematics, science, and engineering. Practice converting between these systems to solidify your understanding and enhance your problem-solving skills.