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Rectangular coordinates, often termed as Cartesian coordinates, are a fundamental concept in mathematics and physics. These coordinates help pinpoint the location of a point within a plane, using two numerical values. Typically, these values are denoted as \(x, y\). The \(x\) value describes a point's position horizontally, while the \(y\) value specifies its vertical position. For example, the coordinates (-1, 1) indicate:

• Move one unit horizontally left from the origin, because the value -1 is negative.
• Move one unit vertically up from the origin, because the value 1 is positive.

Combining these steps places the point in the second quadrant of the Cartesian plane. Hence, the rectangular coordinates offer a straightforward method to locate any point on a 2D plane by accounting for both horizontal and vertical distances.

The Cartesian plane, named after René Descartes, is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It's a foundational graphing system where every point is expressed as an ordered pair \(x, y\).

The plane is divided into four sections called quadrants:

• First Quadrant: Both \(x\) and \(y\) are positive.
• Second Quadrant: \(x\) is negative, and \(y\) is positive.
• Third Quadrant: Both \(x\) and \(y\) are negative.
• Fourth Quadrant: \(x\) is positive, and \(y\) is negative.

Given the example point (-1,1), it's situated in the second quadrant. Here, the x-coordinate is negative and the y-coordinate is positive, depicting that we go left on the x-axis and up on the y-axis from the origin. This coordinate system is pivotal in analytical geometry, facilitating the representation and analysis of geometric figures.

Angle conversion is essential when transitioning between rectangular and polar coordinates. Polar coordinates typically include a radius \(r\) and an angle \(\theta\). The process involves calculating these values from the given \(x\) and \(y\) in the Cartesian system.

To obtain the radius \(r\), use the formula:\[r = \sqrt{x^2 + y^2}\]This computes the distance from the point to the origin. For the given point (-1,1), \(r\) becomes \((\sqrt{2})\).

Determining the angle \(\theta\) involves the arctangent function:\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]Due to the nature of angle positioning in different quadrants, adjustments are necessary. For instance, a point in the second quadrant like (-1,1) requires adding \(180^\circ\) to account for its true position, resulting in an angle of \(135^\circ\). This understanding is crucial for accurately representing points in polar coordinates, offering a distinct perspective on their position within the plane.