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Chapter 6: Q 6.3-6E (page 306)

Question: What is the relationship between the volumes of the tetrahedron defined by the vectors

and the area of the triangle with vertices

See Exercises 4 and 5. Explain this relationship geometrically. Hint: Consider the top face of the tetrahedron.

Short Answer

Expert verified

Therefore, the relationship between volume of tetrahedron and area of triangle is given by,

V2 = 3V1.

Step by step solution

01

Definition. 

Area of the triangle:

Basically, it is equal to half of the Area of the parallelogram.

The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle.

A = 1/2 脳 b 脳 h

Area of tetrahedrons:

It is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

It is also known as a triangular pyramid.

General formula of the volume of the tetrahedrons is

V = 1/3 ( Area of base ) ( perpendicular height )

02

To find the relationship between volume of tetrahedron and area of triangle.

From what we know from ex. 5, the area of a tetrahedron defined by

is,

On the other hand, the area of a triangle with vertices

is

Therefore

V2= 3V1

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Most popular questions from this chapter

29. IfA=u鈫�v鈫�w鈫�is a3脳3 matrix, then the formuladetA=V鈫�.u鈫�脳w鈫� must hold.

Consider the quadrilateral in the accompanying figure, with vertices Pi=(xi,yi), for i=1,2,3,4. Show that the area of this quadrilateral is

12(det[x1x2y1y2]+det[x2x3y2y3]+det[x3x4y3y4]+det[x4x1y4y1]).

Use Exercise 31 to find

det[1111112345149162518276412511681256625]

Do not use technology.

There exists an invertible matrix of the form

[aefjb0g0c0h0d0i0]

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

3.[13241648130026412]
See all solutions

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