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Rectangular coordinates are a way to locate points on a plane using two values, commonly known as the x-coordinate and the y-coordinate. This system is also known as the Cartesian coordinate system. Imagine a grid that extends infinitely in both directions; this is essentially what the rectangular coordinate system looks like.
Each point on the grid is described by two numbers. For instance, in the point (2,0), "2" is the x-coordinate meaning it is two units away from the origin along the horizontal axis, and "0" is the y-coordinate meaning it is directly on the x-axis.

• The x-coordinate indicates horizontal position.
• The y-coordinate indicates vertical position.

The vertical line in rectangular coordinates through the point (2,0) would mean that every point on this line has an x-coordinate of 2, hence the equation x = 2. Rectangular coordinates are particularly simple when dealing with vertical or horizontal lines because one of the coordinates remains constant.

Polar coordinates, in contrast to rectangular coordinates, use a different system to pinpoint a location on the plane. Instead of x and y values, polar coordinates use the radius and angle, often denoted as \(r\) and \(\theta\) respectively.
Here's what these mean:

• \(r\) represents the distance from the origin to the point.
• \(\theta\) represents the angle formed with the positive x-axis.

To convert from rectangular to polar coordinates, you use these key relationships:

• \(x = r \cdot \cos(\theta)\)
• \(y = r \cdot \sin(\theta)\)

For the line x = 2, we convert it using the formula for x in polar coordinates: \(x = r \cdot \cos(\theta)\). Setting \(x\) equal to 2, the polar equation becomes \(r \cdot \cos(\theta) = 2\). Polar coordinates are particularly useful in contexts involving circular shapes or rotations, providing an intuitive way to describe curves, like circles or spirals.

A vertical line is a straight path that goes straight up and down, parallel to the y-axis. When translated into equations, vertical lines have a special feature: they are described by an equation like \(x = a\), where \(a\) is a constant number. In the example given with the point (2,0), the vertical line has an equation of \(x = 2\).
This trait stems from all points on such a line sharing the same x-coordinate. No matter how high or low you go along the line, the x-value is unchanging. In rectangular coordinates, vertical lines are easy to recognize because their equations only involve the x-variable.
When converting to polar coordinates, identifying a vertical line involves expressing the invariant nature of the x-coordinate in terms of \(r\) and \(\theta\). As seen before, the equation \(r \cdot \cos(\theta) = 2\) encapsulates this idea in the polar system.