Question

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

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

V₂ = 3V_1.

Step-by-step solution

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 × b × 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 )

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