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Chapter 11: Problem 81

THINK ABOUT IT What is the \(z\)-coordinate of any point in the \(xy\)-plane? What is the \(y\)-coordinate of any point in the \(xz\)-plane? What is the \(x\)-coordinate of any point in the \(yz\)-plane?

Short Answer

Expert verified
The \(z\)-coordinate of any point in the \(xy\)-plane is 0. The \(y\)-coordinate of any point in the \(xz\)-plane is 0. The \(x\)-coordinate of any point in the \(yz\)-plane is 0.

Step by step solution

01

Understand the xy-plane

The xy-plane is a horizontal plane in 3-dimensional space where all points on this plane have a \(z\)-coordinate of 0, irrespective of the values of \(x\) and \(y\) coordinates.
02

Understand the xz-plane

The xz-plane, on the other hand, is a vertical plane that includes the x and z-axes. All points on this plane have a \(y\)-coordinate of 0, no matter what the \(x\) and \(z\) coordinates are.
03

Understand the yz-plane

Similar to the xz-plane, the yz-plane is also a vertical plane but it includes the y and z-axes. Hence, any point on the yz-plane will have an \(x\)-coordinate of 0, regardless of the \(y\) and \(z\) values.

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Key Concepts

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

xy-plane
In the world of mathematics, particularly geometry, understanding the concept of the xy-plane helps students navigate the complexities of three-dimensional space effectively. The xy-plane is one of the main coordinate planes in three-dimensional space. It's often easiest to think of the xy-plane as a flat surface similar to the floor you're standing on.

  • In this plane, the third dimension, which is the z-coordinate, is always 0.
  • This means every point in this plane has coordinates in the form (x, y, 0).
  • The x and y coordinates can vary freely, covering any combination of horizontal and vertical positions.
By imagining a piece of paper lying flat with the edges aligned with the x and y axes, you will grasp the concept of the xy-plane. It serves as a base map for movements in two-dimensional space while remaining fixed in the three-dimensional world.
xz-plane
The xz-plane is an integral part of understanding vertical planes within a 3D coordinate system. Just like other planes, all points along this plane follow a specific set of rules that govern their position.

  • The key to recognizing points on the xz-plane is that the y-coordinate is always 0.
  • Any point located here can be represented with coordinates (x, 0, z). This highlights the absence of any movement along the y-axis.
  • The x and z values can be any number, positioning points anywhere along these axes.
Visualizing the xz-plane is as if you are standing facing a wall, where you can move up and down (along the z-axis) or left and right (along the x-axis) but cannot move in or out towards you in the space, which belongs to the y dimension.
yz-plane
Comprehending the yz-plane is often the final puzzle piece in understanding 3D planes. The yz-plane holds a significant role as it intersects the y and z axes, leaving particular characteristics for the coordinates of points on this plane.

  • For all points on the yz-plane, the x-coordinate is fixed at 0. This essentially means there is no motion along the x-axis.
  • Coordinates for points are written as (0, y, z), signifying that only the y and z coordinates can vary.
  • This stability along the x-axis allows movement horizontally and vertically along a vertical plane.
Picture standing alongside a tall bookshelf where you can only pick books up and down the shelving (movement along z) or lateral movement along the shelf (along y), this gives a fitting illustration of the yz-plane. It helps in guiding you through the framework of 3D geometry.

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Most popular questions from this chapter

THINK ABOUT IT If the magnitudes of two vectors are doubled, how will the magnitude of the cross product of the vectors change?

In Exercises 21-30, find \(\textbf{u} \times \textbf{v}\) and show that it is orthogonal to both \(\textbf{u}\) and \(\textbf{v}\). \(\textbf{u} = 6\textbf{i}+2\textbf{j}+\textbf{k}\) \(\textbf{v} = \textbf{i}+3\textbf{j}-2\textbf{k}\)

In Exercises 31-36, find the general form of the equation of the plane with the given characteristics. Passes through \((1, -2, 4)\) and \((4, 0, -1)\) and is perpendicular to the \(xz\)-plane

In Exercises 37-42, find the area of the parallelogram that has the vectors as adjacent sides. \(\textbf{u} = \textbf{i}+2\textbf{j}+2\textbf{k}\) \(\textbf{v} = \textbf{i}+\textbf{k}\)

In Exercises 21-30, find \(\textbf{u} \times \textbf{v}\) and show that it is orthogonal to both \(\textbf{u}\) and \(\textbf{v}\). \(\textbf{u} = \langle 6, 8, 3 \rangle\) \(\textbf{v} = \langle 5, -2, -5 \rangle\)
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