What representations do not change the location of the given point (7,140°)?
To find the representations that do not change the location of the given point (7, 140°), we need to consider the different ways a point can be represented in mathematics. The two common representations are Cartesian coordinates (x, y) and polar coordinates (r, θ).
1. Cartesian Coordinates: In Cartesian coordinates, the given point (7, 140°) is represented as (x, y), where x denotes the horizontal distance and y represents the vertical distance. So, any representation that keeps the values of x and y the same will not change the location of the point.
2. Polar Coordinates: In polar coordinates, the given point (7, 140°) is represented as (r, θ), where r denotes the distance from the origin (0, 0) to the point and θ represents the angle between the positive x-axis and the line connecting the origin to the point (measured counterclockwise). To keep the location of the point unchanged, we need to ensure that the distance (r) and the angle (θ) remain the same.
Based on these representations, the representations that do not change the location of the given point (7, 140°) are:
- Cartesian Coordinates: Any point in the form (7, y), where y can be any real number, will keep the location of the point unchanged.
- Polar Coordinates: Any point in the form (7, 140° + 360°n), where n is an integer, will keep the location of the point unchanged.
To identify representations that do not change the location of a given point (7,140°), we need to consider different coordinate systems and transformations.
1. Cartesian Coordinates: The point (7,140°) in Cartesian coordinates represents a point in the plane. Since the point is given as polar coordinates, we need to convert it to Cartesian coordinates. The conversion formula is as follows:
x = r * cos(θ)
y = r * sin(θ)
where r is the magnitude or radius, and θ is the angle in radians or degrees. In this case, r = 7 and θ = 140°.
Calculating the Cartesian coordinates:
x = 7 * cos(140°)
y = 7 * sin(140°)
Any representations of the point in (x, y) coordinates that satisfy these equations will not change the location of the point.
2. Polar Coordinates: Since the point is already given in polar coordinates, any representation with the same magnitude and angle will keep the point in the same location. Therefore, any other polar representation of (7,140°), such as (7,-220°) or (7,500°), will not change the location of the point.
3. Vector Representation: A vector representation of a point can also be used. In this case, we can use the magnitude and direction of the vector to represent the point. The vector component notation is:
⟨x, y⟩
Converting the polar coordinates to vector representation:
x = 7 * cos(140°)
y = 7 * sin(140°)
The vector representation of the point (7,140°) is ⟨7 * cos(140°), 7 * sin(140°)⟩.
In summary, representations that do not change the location of the given point (7,140°) include Cartesian coordinates (x, y) where x = 7 * cos(140°) and y = 7 * sin(140°), as well as any other polar coordinates with the same magnitude and angle, such as (7,-220°) or (7,500°). Additionally, the point can be represented as a vector ⟨7 * cos(140°), 7 * sin(140°)⟩.