Problem 18 Draw the $x y$ region $R$ th... [FREE SOLUTION]

Chapter 14: Problem 18

Draw the $x y$ region $R$ that corresponds to the $u v$ square $S$ with corners (0,0),(1,0),(0,1),(1,1) . Locate the corners of $R$ and then its sides (bike a jigsaw puzzle). $$ x=3 u+2 v, y=u+v $$

Short Answer

Expert verified

Region $ R $ in the $ xy $-plane is a quadrilateral with corners (0,0), (3,1), (2,1), and (5,2).

Step by step solution

Identify the Corners of Square S in the uv-plane

The given square $ S $ in the $ uv $-plane has corners at $(0, 0), (1, 0), (0, 1), (1, 1)$. These points will be transformed to new points in the $ xy $-plane using the provided transformation equations.

Transform Corner (0, 0)

Using the transformation equations $ x = 3u + 2v $ and $ y = u + v $, for the corner $(0, 0)$:- $ x = 3(0) + 2(0) = 0 $- $ y = 0 + 0 = 0 $Thus, this corner transforms to $(0, 0)$ in the $ xy $-plane.

Transform Corner (1, 0)

For the corner $(1, 0)$:- $ x = 3(1) + 2(0) = 3 $- $ y = 1 + 0 = 1 $Thus, this corner transforms to $(3, 1)$ in the $ xy $-plane.

Transform Corner (0, 1)

For the corner $(0, 1)$:- $ x = 3(0) + 2(1) = 2 $- $ y = 0 + 1 = 1 $Thus, this corner transforms to $(2, 1)$ in the $ xy $-plane.

Transform Corner (1, 1)

For the corner $(1, 1)$:- $ x = 3(1) + 2(1) = 5 $- $ y = 1 + 1 = 2 $Thus, this corner transforms to $(5, 2)$ in the $ xy $-plane.

Identify the Region R in the xy-plane

The transformed corners in the $ xy $-plane are $(0, 0)$, $(3, 1)$, $(2, 1)$, and $(5, 2)$. These points form a quadrilateral in the $ xy $-plane, which is the region $ R $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transformation of Coordinates

In geometry, a transformation of coordinates involves changing the reference system in which geometric figures are described. In this exercise, we use coordinate transformations to describe a shape, the square, from one plane to another. Here, we are transforming a square from the "uv-plane" to the "xy-plane" using specific equations:

First, we have the transformation equations: $ x = 3u + 2v $ and $ y = u + v $. These determine how each point in the "uv-plane" maps into a point in the "xy-plane".
Coordinate transformations help in understanding how geometrical entities like points, lines, and, in this case, a square, change their position and orientation during a transformation.

Being familiar with these transformations is crucial in fields like computer graphics, physics, and engineering, where geometric representations often need to be altered.

xy-plane

The "xy-plane" is a two-dimensional plane where each point is determined by an x-coordinate and a y-coordinate. In this exercise, the square's corners from the "uv-plane" are transformed to new positions on the "xy-plane".

For example, using the transformations for the corner $(1, 0)$ gives $x = 3 \cdot 1 + 2 \cdot 0 = 3$ and $y = 1 + 0 = 1$, placing this point at $(3, 1)$ on the "xy-plane".
Understanding the role of the "xy-plane" is essential for interpreting how transformations alter the shape's placement, as this plane is a common reference in mathematics and engineering.

The converted coordinates on the "xy-plane" illustrate where and how the initial square's vertices are repositioned after applying the transformation equations.

uv-plane

The "uv-plane" is another two-dimensional coordinate system consisting of axes labeled "u" and "v". It's the original plane in this problem where we define a geometric figure, specifically a square.

This square has corners at points $(0, 0), (1, 0), (0, 1)$, and $(1, 1)$ on the "uv-plane"; these are the inputs into the coordinate transformation equations.
These points are then taken as inputs to obtain new coordinates on the "xy-plane", showing how coordinates in one system can correspond to coordinates in another system.

The exercise essentially demonstrates how a simple shape in the "uv-plane" becomes a more complex figure in another coordinate system.

Quadrilateral

A quadrilateral is a four-sided polygon with four vertices and four edges. In this exercise, the transformation of the coordinates results in the vertices of a quadrilateral on the "xy-plane".

The initial square in the "uv-plane" possesses equal and perpendicular sides, but after transformation, the resulting figure in "xy-plane" typically does not maintain these properties.
After calculating the new vertex positions through transformation, the square's original right-angled nature in the "uv-plane" changes to a quadrilateral with potentially unequal sides and varying angles in the "xy-plane".
This showcases the importance of understanding transformations as they can drastically change the geometry of figures.

Grasping how squares become quadrilaterals through transformations is crucial in fields like graphics design and architecture, where converting between different dimensions and perspectives often occurs.

91影视

Draw the \(x y\) region \(R\) that corresponds to the \(u v\) square \(S\) with corners (0,0),(1,0),(0,1),(1,1) . Locate the corners of \(R\) and then its sides (bike a jigsaw puzzle). $$ x=3 u+2 v, y=u+v $$

Short Answer

Step by step solution

Identify the Corners of Square S in the uv-plane

Transform Corner (0, 0)

Transform Corner (1, 0)

Transform Corner (0, 1)

Transform Corner (1, 1)

Identify the Region R in the xy-plane

Key Concepts

Transformation of Coordinates

xy-plane

uv-plane

Quadrilateral

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