Find the volume of the tetrahedron formed with the following vertices: (1,5,-2) (3,1,4) (0,-1,1) (2,-2,5)?

Question

volume = area of the base * height divided by 3.
the answer is 2/3 units cubed but i don't get how u find that volume. please explain.

just trying to kill time with math problems except i'm stumped.

Answer 1

Find the volume of the tetrahedron formed with the following vertices: P(1,5,-2); Q(3,1,4); R(0,-1,1); and S(2,-2,5).

As you have noted the volume of the tetrahedron is (1/3)bh. So the approach to take is to choose three vertices to be the base and find the area of the base. Then find the equation of the plane thru the three points. Finally find the distance of the fourth point from the plane. That will be the height of the tetrahedron.

First, let's define the base of the tetrahedron with:
P(1,5,-2); Q(3,1,4); R(0,-1,1)

Let
u = PQ = <3-1, 1-5, 4+2> = <2, -4, 6>
v = PR = <0-1, -1-5, 1+2> = <-1, -6, 3>

Recall that the area of a triangle with sides u, v, and included angle θ is

Area = (1/2)uvsinθ.

Recall also that the magnitude of the cross product of two vectors u and v is:

| u X v | = | u | | v | sinθ

So the area of the triangle defined by
P(1,5,-2); Q(3,1,4); R(0,-1,1) is:

b = Area = (1/2)| u X v |

u X v = <2, -4, 6> X <-1, -6, 3> = <24, -12, -16>

| u X v | = √[24² + (-12)² + (-16)²] = √976 = 4√61

b = Area = (1/2)| u X v | = (1/2)(4√61) = 2√61
__________________

Find the distance d, from the remaining point S(2,-2,5), to the plane defined by the first three points. It will be in the direction of the vector n = u X v.

Remember the point P(1,5,-2) and define the vector PS.
PS = <2-1, -2-5, 5+2> = <1, -7, 7>

Let θ be the angle of the vector PS to the plane PQR. Then:
cosθ = d / |PS| = d / | n |

d = |PS|cosθ

The dot product of PS and n is:
PS • n = |PS| | n | cosθ

cosθ = (PS • n) / (|PS| | n |)

d = |PS|cosθ = (PS • n) / | n |

Plugging in values we have:

d = (PS • n) / | n | = <1, -7, 7> • <24, -12, -16> / (4√61)
d = (24 + 84 - 112) / (4√61) = -4 / (4√61) = -1/√61

We want the absolute value of that d = 1/√61.

The volume of the tetrahedron is:

V = (1/3)bh = (1/3)(2√61)(1/√61) = 2/3

Answer 2

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Answer 3

i bet this is your homework
lol