Part 2 Coordinate Systems

The video introduces coordinate systems, starting with how location is typically shared as two numbers, representing X and Y coordinates relative to a zero point or origin. It then expands to 3D Cartesian coordinates (X, Y, Z) and explains that the focus will be on 2D systems, neglecting the Z-axis for now.

The 2D Cartesian system consists of an X-axis and a Y-axis, intersecting at the origin (0,0). The plane is divided into four quadrants, each defined by the positive or negative signs of X and Y values. Moving right on the X-axis or up on the Y-axis results in positive values, while moving left or down results in negative values. This allows for determining a point's quadrant based on its coordinate signs.

The video demonstrates how to represent a point (e.g., 1, 2) in the Cartesian system, starting from the origin, moving along the X-axis first, then the Y-axis. It clarifies that the X-coordinate represents the distance from the Y-axis, and the Y-coordinate represents the distance from the X-axis. This process also works if you move along the Y-axis first, and then the X-axis.

Polar coordinates offer an alternative way to represent a point using a distance (R) and an angle (theta) from the origin. The angle is measured counter-clockwise from the positive X-axis for positive angles and clockwise for negative angles. Examples are given for plotting points using R and theta.

The section explains how to convert Cartesian coordinates (X, Y) to Polar coordinates (R, theta). A right-angled triangle is formed with X, Y, and R as its sides. R is calculated using the Pythagorean theorem (R = sqrt(X^2 + Y^2)), and theta is calculated using the arctangent function (theta = arctan(Y/X)).

Conversely, converting from Polar coordinates (R, theta) to Cartesian coordinates (X, Y) involves using trigonometric functions: X = R * cos(theta) and Y = R * sin(theta).

A detailed example demonstrates converting the Cartesian point (3, 4) to Polar coordinates. It shows how to graphically represent the point and use the formulas to find R (which is 5) and theta (approximately 53 degrees), located in the first quadrant.

This example addresses converting a point in the second quadrant, (-3, 4), to Polar coordinates. While R remains 5, calculating theta requires an adjustment. First, an angle (phi) is found using the absolute values of X and Y. Then, since the point is in the second quadrant, the actual theta is 180 - phi (approximately 127 degrees).

The conversion for a point in the third quadrant, (-3, -4), is illustrated. Again, R is 5. For theta, after finding phi, it's determined that theta is 180 + phi (approximately 233 degrees).

The final example covers a point in the fourth quadrant, (3, -4). R is still 5. Theta is calculated as 360 - phi (approximately 307 degrees) or alternatively as -phi, indicating a negative angle. The video emphasizes that calculators might not correctly identify the quadrant when solely using arctan, making the manual quadrant-based adjustment crucial.

The video concludes by summarizing the general steps to convert Cartesian (X, Y) to Polar (R, theta): first, calculate R using the Pythagorean theorem. Second, calculate an auxiliary angle (phi) using arctan of the absolute values of Y/X. Finally, determine the true angle theta based on the quadrant of the original Cartesian point: Quadrant 1 (theta = phi), Quadrant 2 (theta = 180 - phi), Quadrant 3 (theta = 180 + phi), or Quadrant 4 (theta = 360 - phi or -phi).