Problem 4
Question
Suppose \(S\) is the unit cube in the first octant of \(u v w\) -space with one vertex at the origin. What is the image of the transformation \(T: x=u / 2, y=v / 2, z=w / 2 ?\)
Step-by-Step Solution
Verified Answer
Answer: The image of the given transformation is a cube with side length 1/2, located in the first octant of xyz-space.
1Step 1: Identify the vertices of the unit cube S
The unit cube S has 8 vertices, which are (0,0,0), (1,0,0), (1,1,0), (0,1,0), (0,0,1), (1,0,1), (1,1,1), and (0,1,1).
2Step 2: Apply the transformation T to each vertex
Apply the transformation \(T: x=u/2, y=v/2, z=w/2\) to each vertex in the unit cube. We will get 8 new points representing the image of the transformation
$$
(0,0,0) \rightarrow(0/2,0/2,0/2)=(0,0,0) \\
(1,0,0) \rightarrow(1/2,0/2,0/2)=(1/2,0,0) \\
(1,1,0) \rightarrow(1/2,1/2,0/2)=(1/2,1/2,0) \\
(0,1,0) \rightarrow(0/2,1/2,0/2)=(0,1/2,0) \\
(0,0,1) \rightarrow(0/2,0/2,1/2)=(0,0,1/2) \\
(1,0,1) \rightarrow(1/2,0/2,1/2)=(1/2,0,1/2) \\
(1,1,1) \rightarrow(1/2,1/2,1/2)=(1/2,1/2,1/2) \\
(0,1,1) \rightarrow(0/2,1/2,1/2)=(0,1/2,1/2)
$$
3Step 3: Determine the shape formed by the transformed vertices
Now, look at the transformed vertices to see what shape they form. The points are (0,0,0), (1/2,0,0), (1/2,1/2,0), (0,1/2,0), (0,0,1/2), (1/2,0,1/2), (1/2,1/2,1/2), and (0,1/2,1/2). These points form a new cube with side length 1/2.
So, the image of the transformation \(T: x=u/2, y=v/2, z=w/2\) is a cube with side length 1/2, also located in the first octant of \(xyz\)-space.
Key Concepts
Linear TransformationUnit CubeThree-Dimensional SpaceCoordinate System
Linear Transformation
A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. In simpler terms, if you have two vectors that are transformed, the sum of their transformations is equal to the transformation of their sum. Similarly, scaling a vector and then transforming it is the same as transforming the vector first and then scaling the result.
Using the example provided, the transformation given by the equations T: x=u/2, y=v/2, z=w/2 is linear because each of the new coordinates (x, y, z) is obtained by simply scaling the original coordinates (u, v, w) by a factor of 1/2. There is no addition of vectors or multiplication by different scalars within the transformation, making this a textbook example of a linear transformation in practice.
Using the example provided, the transformation given by the equations T: x=u/2, y=v/2, z=w/2 is linear because each of the new coordinates (x, y, z) is obtained by simply scaling the original coordinates (u, v, w) by a factor of 1/2. There is no addition of vectors or multiplication by different scalars within the transformation, making this a textbook example of a linear transformation in practice.
Unit Cube
The term unit cube refers to a cube whose edges all have a length of one unit. In three-dimensional space, a unit cube is often used as a building block to visualize complex shapes and transformations.
In our case, the unit cube in question is positioned in the first octant of a uvw coordinate system with one of its vertices at the origin, this is the point (0,0,0). This standard position makes it easier to apply transformations and understand their effects, as seen with transformation T that scales the unit cube by 1/2 along each axis, resulting in a smaller cube that retains the same orientation and proportion.
In our case, the unit cube in question is positioned in the first octant of a uvw coordinate system with one of its vertices at the origin, this is the point (0,0,0). This standard position makes it easier to apply transformations and understand their effects, as seen with transformation T that scales the unit cube by 1/2 along each axis, resulting in a smaller cube that retains the same orientation and proportion.
Three-Dimensional Space
Three-dimensional space is the geometric setting in which three values are required to determine the position of an element (i.e., a point). Unlike two-dimensional space, which has length and width, three-dimensional space also includes height or depth, which allows for a more realistic representation of objects and their relationships.
The concept is crucial for understanding geometric transformations such as the one described in the exercise. After the unit cube undergoes the linear transformation T, we can still visualize and measure its existence in three-dimensional space, observing its new position, orientation, and dimensions.
The concept is crucial for understanding geometric transformations such as the one described in the exercise. After the unit cube undergoes the linear transformation T, we can still visualize and measure its existence in three-dimensional space, observing its new position, orientation, and dimensions.
Coordinate System
A coordinate system is an essential tool for representing points within a space, using numbers to provide the location of a point in one, two, or more dimensions. In this scenario, the uvw and xyz notation indicates the use of a three-dimensional Cartesian coordinate system for each respective space.
The Cartesian coordinate system is composed of three axes, commonly denoted as x, y, and z, that are all perpendicular to each other, dividing space into eight octants. In the unit cube's transformation exercise, the transformation mapping from uvw to xyz is clearly described, illustrating how each point from one system is translated into another using the transformation rule T.
The Cartesian coordinate system is composed of three axes, commonly denoted as x, y, and z, that are all perpendicular to each other, dividing space into eight octants. In the unit cube's transformation exercise, the transformation mapping from uvw to xyz is clearly described, illustrating how each point from one system is translated into another using the transformation rule T.
Other exercises in this chapter
Problem 3
Sketch the region of integration for the integral \(\int_{-\pi / 6}^{\pi / 6} \int_{1 / 2}^{\cos 2 \theta} f(r, \theta) r d r d \theta\)
View solution Problem 3
Which order of integration is preferable to integrate \(f(x, y)=x y\) over \(R=\\{(x, y): y-1 \leq x \leq 1-y, 0 \leq y \leq 1\\} ?\)
View solution Problem 4
In the integral for the moment \(M_{x}\) of a thin plate, why does \(y\) appear in the integrand?
View solution Problem 4
Describe the set \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) in spherical coordinates.
View solution