?Consider points \(A(\alpha, 0,0)\), \(B(0, \beta, 0)\), and \(C(0,0, \gamma)\), with

Chapter 2, Problem 220

(choose chapter or problem)

Consider points \(A(\alpha, 0,0)\), \(B(0, \beta, 0)\), and \(C(0,0, \gamma)\), with \(\alpha\), \(\beta\), and \(\gamma\) positive real numbers.

a. Determine the volume of the parallelepiped with adjacent sides \(\overrightarrow{O A}\) , \(\overrightarrow{O B}\) , and \(\overrightarrow{O C}\) .

b. Find the volume of the tetrahedron with vertices O, A, B, and C. (Hint: The volume of the tetrahedron is 1/6 of the volume of the parallelepiped.)

c. Find the distance from the origin to the plane determined by A, B, and C. Sketch the parallelepiped and tetrahedron.

Text Transcription:

A(alpha, 0,0)

B(0, beta, 0)

C(0,0, gamma)

alpha

beta

gamma

overrightarrow OA

overrightarrow OB

overrightarrow OC