91Ó°ÊÓ

Rectangular coordinates, also known as Cartesian coordinates, are a way to specify points in a 3D space using three components: x, y, and z. The x-component tells you how far along the horizontal axis a point is, while the y-component tells you the extent along the vertical axis in a plane. The z-component reveals the altitude or depth of the point above or below the reference plane. Understanding these coordinates involves visualizing them like directions on a map, where you have to move along the x-axis, and then the y-axis, with the z-axis adding vertical layering. In the given exercise, the point

• x = 2
• y = -2
• z = -4

These components describe the exact location of a point in space relative to a fixed position, known as the origin.

Converting between different coordinate systems involves changing the way we represent a point's location in space. Here, we are converting from rectangular (or Cartesian) coordinates to cylindrical coordinates. The purpose is often to utilize a system that simplifies calculations or better describes the situation we're analyzing.

Cylindrical coordinates combine radial distance (r), angular coordinate (θ), and height (z). To convert, we begin with the rectangular coordinates and compute:

• Radial distance: This is the hypotenuse formed with the x and y components. Using the formula \[ r = \sqrt{x^2 + y^2} \]
• Angular coordinate: This is the angle θ measured from the positive x-axis to the point in the xy-plane. We use \[ \theta = \arctan \left(\frac{y}{x}\right) \]
• Height: This remains the same as the z-coordinate in rectangular coordinates.

Using these formulas, the task is to reinterpret the location given by rectangular coordinates into a cylindrical space view to aid in specific kinds of mathematical or graphical operations.

The polar angle calculation is a critical step when converting from rectangular to cylindrical coordinates as it determines the direction of the point in the plane. The angle \( \theta \) is calculated using the arctan function:
\[ \theta = \arctan \left(\frac{y}{x}\right) \]This function computes the angle whose tangent is the quotient of y and x. If the resultant angle is negative, which is possible when the point ends up in a quadrant that reflects in the negative angle range, you must adjust by adding 360° to ensure the angle is positive and measured in the typical counter-clockwise direction. For example, in the exercise with coordinates (2, -2), the initially calculated angle is -45°. Adding 360° gives a usable angle in cylindrical coordinates of 315°, properly locating our direction on the plane. Ensuring the angle falls within the 0° to 360° range is essential for consistency and accuracy, especially when the angle must align with polar plot conventions.