In the rectangular coordinate system, each point (x, y) has a unique representation. Explain why this is not true for a point (r, Θ) in the polar coordinate system.
the angle can be rotate, past the origin. So theta=3PI is the same ray point as PI.
Ah, the polar coordinate system, where things take a twist! Well, my friend, the reason why each point (r, Θ) in the polar coordinate system may not have a unique representation is because of those sneaky angles.
You see, in the polar coordinate system, the angle Θ can be expressed in multiple ways. For example, if you rotate a point by 360 degrees, you end up in the same position, but with a different angle representation. So, you can have (r, Θ) and (r, Θ + 360°) representing the same point.
But wait, there's more! Sometimes, you might encounter negative angles in the polar coordinate system. So, (r, Θ) and (r, Θ - 180°) could also lead you to the same point.
Basically, in the polar coordinate system, the angles are like clowns doing tricks. They can go in circles, be positive or negative, and make things a bit confusing. So, be prepared for some multiple representations when dealing with points in (r, Θ) form. It's all part of the circus fun!
In the rectangular coordinate system, each point (x, y) has a unique representation because the coordinates x and y represent the horizontal and vertical distances from the origin, respectively. If two points have different x and/or y coordinates, they are considered different points.
However, this is not the case in the polar coordinate system for a point represented as (r, Θ). In polar coordinates, the point (r, Θ) represents a distance r from the origin (known as the radial component) and an angle Θ from the positive x-axis (known as the angular component).
The uniqueness of representation is not guaranteed in polar coordinates because multiple values of r and Θ can correspond to the same physical location or point. For example, if we consider a point with radius r and angle Θ, we can represent the exact same point by adding or subtracting any multiple of 2π to the angle. Similarly, we can represent the same point by considering the negative value of radius r.
Additionally, for points that are on the positive x-axis (where Θ = 0 or 2π), there is an ambiguity in the polar representation. Mathematically, any value of r with Θ equal to 0 or 2π can represent the same point on the positive x-axis.
So, unlike the rectangular coordinate system where each point has a unique representation, the polar coordinate system allows for multiple representations for a given point, making the representation in the polar coordinate system not uniquely determined.
In the rectangular coordinate system, each point (x, y) has a unique representation because the coordinates x and y give the exact position of the point on a two-dimensional plane. This means that no other point can have the same x and y values.
However, in the polar coordinate system, a point (r, Θ) represents a distance r from the origin and an angle Θ from the positive x-axis. And here's where it gets interesting – the same point in the polar coordinate system can be represented by multiple (r, Θ) pairs.
To understand why, imagine pointing at the same spot on a clock with an hour hand that moves continuously. You could be pointing at the same location at 3 o'clock, but you could also represent that spot as both 15 minutes past 2 and 45 minutes before 4. Similarly, in the polar coordinate system, a point can have multiple representations due to the periodic nature of angles.
For example, if we have a point represented by (r, Θ) = (5, 30°), it is also represented by (5, 390°) because adding or subtracting any multiple of 360° gives us an equivalent representation. This means that the point (r, Θ) in polar coordinates does not have a unique representation.
Therefore, to ensure uniqueness, it is common to restrict the angle Θ to a specific range, such as 0° ≤ Θ < 360° or -180° ≤ Θ < 180°. By doing this, we can have a unique representation for each point in the polar coordinate system.