Volume of tetrahedron when you are given four planes

[borovecm]

Homework Statement

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

Homework Equations

\vec{a}=\vec{AB}=(X2-X1)\vec{i}+(y2-y1)\vec{j}+(z2-z1)\vec{k}
\vec{b}=\vec{AC}=(X2-X1)\vec{i}+(y2-y1)\vec{j}+(z2-z1)\vec{k}
\vec{c}=\vec{AD}=(X2-X1)\vec{i}+(y2-y1)\vec{j}+(z2-z1)\vec{k}
V(parallelepiped)=\vec{a}\ast(\vec{b}\times\vec{c})
V(tetrahedron)=1/6*V(parallelepiped)

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 \vec{a}\ast(\vec{b}\times\vec{c}) 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.

 

[HallsofIvy]

I can't tell you where you would use B= B*v/2 since you haven't said what B or v mean in that formula!

 

[borovecm]

Sorry. That's Croatian notation. I think american would be Volume=1/3*B*h where B is area of the base and h is height of tetrahedron. I can calculate h from formula for distance between point where first three planes intersect and the fourth plane. I don't know how to calculate area of the base. Is it correct that volume of this tetrahedron is 2/3?

 

[HallsofIvy]

You can choose any 3 of the 4 vertices to be a triangular base. A quick way of finding the area is to construct vectors \vec{u} and \vec{v} from one of the vertices to the other two. Then the area of the base, the triangle, is B= (1/2)|\vec{u}\times\vec{v}|. The height of the distance from the fourth point to the plane defined by the first three points.

 

[pradyukrish]

You can find out the volume by this formula but it is difficult to calculate the determinant of a 4*4 matrix

 

[LCKurtz]

@borovcm Don't put tex tags around every expression. Type whatever equations would nicely fit on one line and put the tags around that.