Problem 11 - Advanced Engineering Mathematics

Question

Give the coordinates of the vertices of the rectangular parallelepiped whose sides are the coordinate planes and the planes \(x=2, y=5, z=8\)

Step-by-Step Solution

Answer

The vertices are (0,0,0), (2,0,0), (0,5,0), (2,5,0), (0,0,8), (2,0,8), (0,5,8), and (2,5,8).

Step 1: Understand the problem

We are given a parallelepiped that is bounded by the coordinate planes, which are the xy-plane, yz-plane, and zx-plane, and the planes given by the equations \(x=2\), \(y=5\), and \(z=8\). Our task is to find the coordinates of the vertices of this parallelepiped.

Step 2: Identify the corners formed by coordinate planes

The vertex formed by the origin, where all coordinates are zero, is at \((0,0,0)\). This point is one corner of the parallelepiped.

Step 3: Determine vertices formed by intersections with 'x=2'

The plane \(x=2\) intersects the coordinate planes at three points: the xy-plane at \((2, 0, 0)\), the yz-plane at \((2, 5, 0)\), and the zx-plane at \((2, 0, 8)\). These are three vertices of the parallelepiped.

Step 4: Determine vertices formed by intersections with 'y=5'

The plane \(y=5\) intersects the coordinate planes at: the xy-plane at \((0, 5, 0)\), the yz-plane at \((0, 5, 8)\), and the zx-plane intersects at \((2, 5, 0)\), which we have already considered. These are additional vertices.

Step 5: Determine vertices formed by intersections with 'z=8'

The plane \(z=8\) intersects the coordinate planes at three points: the xy-plane at \((0, 0, 8)\), the yz-plane at \((0, 5, 8)\), and the zx-plane at \((2, 0, 8)\), which we have already considered. We now have all vertices.

Step 6: List all unique vertices

The vertices of the parallelepiped are: \((0,0,0)\), \((2,0,0)\), \((0,5,0)\), \((2,5,0)\), \((0,0,8)\), \((2,0,8)\), \((0,5,8)\), and \((2,5,8)\).

Key Concepts

Coordinate Planes

In the world of geometry, coordinate planes serve as fundamental building blocks for understanding 3D shapes. The coordinate planes in a 3D system are the planes that divide space into distinct regions. Each plane corresponds to a combination of two of the x, y, and z axes. These include:

• The xy-plane, where z is zero.
• The yz-plane, where x is zero.
• The zx-plane, where y is zero.

Coordinate planes provide a reference framework that helps to locate points in space. They are essential for visualizing geometric figures and analyzing intersections of various planes.

Vertices Identification

Vertices are pivotal points in geometry where two or more edges meet. In the context of a rectangular parallelepiped, these are the points where the faces of the shape intersect. For example, in a rectangular parallelepiped bounded by planes like x=2, y=5, and z=8, and the coordinate planes, vertices are formed at the intersections:

• When x, y, and z values are zero, we have the vertex at (0, 0, 0).
• Each additional plane adds specific coordinates forming more vertices such as (2, 0, 0) or (0, 5, 0).

Identifying vertices involves finding these intersection points.
Vertices are crucial because they help determine the overall shape and dimensions of solids.

Geometry of Solids

Geometry of solids pertains to the 3D characteristics of figures such as volume, surface area, and shape. A rectangular parallelepiped is a six-faced 3D shape, also known as a box:

• All its faces are rectangles.
• Opposite faces are equal and parallel.
• It has 8 vertices, 12 edges, and 6 faces.

This understanding is vital when calculating properties like volume, which is found by multiplying its length, width, and height. For instance, if the dimensions of the parallelepiped are 2, 5, and 8, its volume would be: \[ Volume = 2 \times 5 \times 8 = 80 \]Additionally, the surface area can be calculated by summing up the areas of all six rectangles that form the shape.

3D Coordinate System

A 3D coordinate system allows us to identify positions in space using three values. These values are typically represented by (x, y, z) coordinates, which determine a point's location along the x, y, and z axes respectively.

• The x-axis runs horizontally.
• The y-axis runs vertically.
• The z-axis adds depth.

In this system, we can precisely locate any point, and it is particularly useful for solving problems related to 3D shapes, like the placement of a rectangular parallelepiped.
Understanding this system is fundamental for visualizing and solving geometrical problems in three dimensions. It aids in determining the spatial relationships between different elements such as points, lines, and planes.