Problem 5 - Calculus Early Transcendentals

Question

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=2 u, y=v / 2$$

Step-by-Step Solution

Answer

Answer: The image of the unit square S under the transformation T is a rectangle in the xy-plane with vertices (0,0), (2,0), (0,0.5), and (2,0.5).

Step 1: Identify the vertices of the unit square S

The unit square S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} can be represented as follows: - Vertex A: (0, 0) - Vertex B: (1, 0) - Vertex C: (1, 1) - Vertex D: (0, 1)

Step 2: Apply the transformation T to find the coordinates of the new vertices in the xy-plane

Now, apply the given transformation T to each vertex: - Vertex A': x = 2(0), y = (0)/2, which is (0,0) - Vertex B': x = 2(1), y = (0)/2, which is (2,0) - Vertex C': x = 2(1), y = (1)/2, which is (2,0.5) - Vertex D': x = 2(0), y = (1)/2, which is (0,0.5)

Step 3: Determine the image of S in the xy-plane and provide its description

The new coordinates for vertices A', B', C', and D' are (0,0), (2,0), (2,0.5), and (0,0.5), respectively. This set of vertices represents a rectangle in the xy-plane with sides parallel to the x and y-axis. The length is 2 (going from A' to B') and the width is 0.5 (going from A' to D'). Therefore, the image of the unit square S under the transformation T is a rectangle in the xy-plane with vertices (0,0), (2,0), (0,0.5), and (2,0.5).

Key Concepts

Transformation of Figures

Understanding the transformation of figures is crucial when dealing with changes in the geometry of shapes on the coordinate plane. A transformation is an operation that moves or changes a shape in some way to produce a new figure, referred to as the 'image'. There are several types of transformations, including translations (sliding), rotations (turning), reflections (flipping), and scaling (resizing).

For instance, our exercise mentions a specific transformation defined by the formulae:

• \(x=2u\)
• \(y=\frac{v}{2}\)

This is a scaling transformation where the figure is stretched horizontally by a factor of 2 and compressed vertically by a factor of 0.5. To visualize this, imagine pulling the sides of a square outwards to form a rectangle while simultaneously pressing its top and bottom towards the center.

It's essential to systematically apply the given transformation to each vertex of the original figure. By doing so, we can precisely determine the position and orientation of the figure's image on the coordinate plane, maintaining the geometrical relationships between points.

Mapping in the Coordinate Plane

Mapping in the coordinate plane is the process of determining the locations of points after a transformation has been applied. It is akin to plotting a route on a map, ensuring each point moves according to the rules of the transformation. In our exercise, the transformation process is a mapping from the uv-plane to the xy-plane.

Here’s how the mapping works step-by-step:

• Start with the original coordinates of each vertex in the uv-plane.
• Use the transformation equations to calculate the new x and y coordinates for each vertex.
• Plot the new vertices on the xy-plane.
• Connect the vertices in the xy-plane to visualize the figure's image.

This mapping of vertices helps us create an equivalent figure in the xy-plane that corresponds to the original figure in the uv-plane. Moreover, it is essential to maintain the order of vertices when connecting them to ensure the figure’s shape is preserved accurately.

Vertices of a Square

A square is a fundamental geometric shape with four equal sides and four right angles. What defines a square in the coordinate plane are its vertices—the corner points. Typically, if we have a unit square positioned with one vertex at the origin of a coordinate system, we can denote its vertices as follows:

• Vertex A at \((0, 0)\)
• Vertex B at \((1, 0)\)
• Vertex C at \((1, 1)\)
• Vertex D at \((0, 1)\)

These vertices give us the necessary information to determine the location of all four sides, making up our square. In coordinate transformation exercises like the one we're discussing, the vertices serve as key anchor points for applying the transformation equations. By calculating the image of each vertex and plotting them, we can accurately depict the transformed figure, be it another square, a rectangle, or any polygon, depending on the nature of the transformation applied.