91Ó°ÊÓ

Consider the cube with edges of length 2 running parallel to the coordinate axes, with its centre at the origin \((0,0,0),\) and with opposite corners at (1,1,1) and (-1,-1,-1) . The \(x-, y-,\) and \(z\) -axes, and the \(x y\) -, \(y z-,\) and \(z x\) -planes cut this cube into eight unit cubes \(-\) one sitting in each octant. (i) Let \(A=(0,0,1), B=(1,0,0), C=(0,1,0), W=(1,1,1) .\) Describe the solid \(A B C W\) (ii) Let \(D=(-1,0,0), X=(-1,1,1) .\) Describe the solid \(A C D X\). (iii) Let \(E=(0,-1,0), Y=(-1,-1,1) .\) Describe the solid \(A D E Y\). (iv) Let \(Z=(1,-1,1) .\) Describe the solid \(A E B Z\). (v) Let \(F=(0,0,-1)\) and repeat steps (i)-(iv) to obtain the four mirror image solids which lie beneath the \(x y\) -plane. (vi) Describe the solid \(A B C D E F\) which is surrounded by the eight identical solids in (i)-(v).

Step by step solution

01

Understanding Octant Splits

The cube is centered at the origin with lengths of 2 units on each edge, giving opposite vertices at (1,1,1) and (-1,-1,-1). This setup ensures that the cube is cut into 8 smaller cubes by the x-, y-, and z-axes, each with a dimension of 1 unit.

02

Describe Solid ABCW

For the solid ABCW, vertices A, B, C, and W form a tetrahedron. These vertices are at coordinates (0,0,1), (1,0,0), (0,1,0), and (1,1,1), respectively. This solid occupies a portion above the xy-plane bounded by these points.

03

Describe Solid ACDX

The solid ACDX is a tetrahedron with vertices A (0,0,1), C (0,1,0), D (-1,0,0), and X (-1,1,1). This shape lies in the upper left octant and extends partially along negative x-axis.

04

Describe Solid ADEY

The solid ADEY is a tetrahedron with vertices A (0,0,1), D (-1,0,0), E (0,-1,0), and Y (-1,-1,1). This construct resides in the lower left octant (in terms of y-axis) and heads into negatives for both x- and y-coordinates.

05

Describe Solid AEBZ

The tetrahedron AEBZ has vertices A (0,0,1), E (0,-1,0), B (1,0,0), and Z (1,-1,1). This solid exists in the lower right xy-octant with respect to the y-axis and extends into positive x and negative y directions.

06

Describe Mirror Image Solids

For steps (i)-(iv) mirrored across the xy-plane, use F (0,0,-1) to create counterparts below the xy-plane. The process is similar, substituting A with F for solids like FBCW, FCDX, FDEY, and FEBZ.

07

Describe Solid ABCDEF

Solid ABCDEF is a hexagonal prism central solid, bound between the eight tetrahedrons. It's the area trapped centrally between planes determined by the tetrahedrons' vertices from (i) to (v).

08

Overall Description Completion

Completing these descriptions outlines the composition of the cube by distinct solids. Each tetrahedral component is distinct, formed by initial vertices. The core is the described hexagonal prism overlooked by the identified sections.

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Key Concepts

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

Tetrahedron

A tetrahedron is one of the simplest three-dimensional shapes. It is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. Each face of a tetrahedron is a triangle, and all vertices are connected directly by edges. This structure makes it one of the fundamental building blocks in geometry. In our cube example, solids like ABCW are tetrahedra. The vertices are distinct points like A (0,0,1), B (1,0,0), C (0,1,0), and W (1,1,1).
These vertices create a spatial figure fitting neatly within a confined section of space in the cube.
This makes understanding a tetrahedron crucial because it helps segment complex shapes into manageable parts.

Coordinate System

A coordinate system is a method to describe positions in a multi-dimensional space using numbers. In a three-dimensional coordinate system, each point is expressed using three values: \(x\), \(y\), and \(z\). These coordinates specify distances along three perpendicular axes: the horizontal \(x\)-axis, the vertical \(y\)-axis, and the depth-related \(z\)-axis. With the cube centered at the origin \((0,0,0)\), vertices are situated equally in all \(x, y, z\) directions.
This system is very useful in geometry for calculating distances and defining shapes. By understanding the distances and spatial relationships, a multitude of shapes like cubes, tetrahedrons, and other polygons can be identified and described efficiently.

Three-Dimensional Shapes

Three-dimensional shapes exist in a space with depth, width, and height. Unlike two-dimensional shapes that lie flat on a plane, 3D shapes can be viewed from multiple perspectives and hold volume. The cube in our exercise is a perfect example of a 3D shape. It's defined by its vertices, edges, and faces within the coordinate space. Other 3D shapes, like the tetrahedron, share the space within the cube. Understanding these shapes involves comprehending how they fit together spatially.

• The cube has 8 vertices, 12 edges, and 6 faces, all equal in size since it's a regular cube.
• Tetrahedrons have 4 vertices and 4 triangular faces, reflecting their compact, simple structure that fills space between the cube's partitions.

Recognizing how these shapes interact is key to solving geometric problems in 3D.

Octants

In a three-dimensional coordinate system, space is divided into eight sections called octants. These are like the quadrants in a two-dimensional system but one step further. The division is defined by the x, y, and z axes as they intersect at the origin. Each octant encompasses a specific part of space with unique coordinate sign combinations. For example, the octant where the solid ABCW resides has all positive coordinates.
The presence of these octants allows for better localization and description of where shapes like our tetrahedrons exist within the cube. Understanding which octant a shape resides in helps when setting bounds for calculations and determining positional properties of three-dimensional solids. This division forms an essential part of understanding spatial arrangements in geometry.