(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have to find volume of tetrahedron that is bounded between 4 planes.

Planes are

x+y+z-1=0

x-y-1=0

x-z-1=0

z-2=0

2. Relevant equations

[tex]\vec{a}[/tex]=[tex]\vec{AB}[/tex]=(X2-X1)[tex]\vec{i}[/tex]+(y2-y1)[tex]\vec{j}[/tex]+(z2-z1)[tex]\vec{k}[/tex]

[tex]\vec{b}[/tex]=[tex]\vec{AC}[/tex]=(X2-X1)[tex]\vec{i}[/tex]+(y2-y1)[tex]\vec{j}[/tex]+(z2-z1)[tex]\vec{k}[/tex]

[tex]\vec{c}[/tex]=[tex]\vec{AD}[/tex]=(X2-X1)[tex]\vec{i}[/tex]+(y2-y1)[tex]\vec{j}[/tex]+(z2-z1)[tex]\vec{k}[/tex]

V(parallelepiped)=[tex]\vec{a}[/tex][tex]\ast[/tex]([tex]\vec{b}[/tex][tex]\times[/tex][tex]\vec{c}[/tex])

V(tetrahedron)=1/6*V(parallelepiped)

3. The attempt at a solution

I found four points where planes meet. These are:

A(1,0,0)

B(0,-1,2)

C(3,2,2)

D(3,-4,2)

From that I made vectors AB, AC, AD and then I put that into [tex]\vec{a}[/tex][tex]\ast[/tex]([tex]\vec{b}[/tex][tex]\times[/tex][tex]\vec{c}[/tex]) and got that volume of parallelepiped is 4. From there I got that volume of this tetrahedron is 2/3. Is this the correct and shortest way to get a solution? My teacher said that I can use formula V=B*v/3 but I don't know where to use it.

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# Homework Help: Volume of tetrahedron when you are given four planes

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