A tetrahedron (triangular pyramid) has vertices (x1, y1, z1),(x2, y2, z2),(x3, y3, z3) and (x4,

step 1 This problem is an application of evaluating a 4x4 determinant. In this problem, you are asked t

A tetrahedron (triangular pyramid) has vertices (x1, y1,...

A tetrahedron (triangular pyramid) has vertices $\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right),\left(x_{3}, y_{3}, z_{3}\right)$ and $\left(x_{4}, y_{4}, z_{4}\right)$. The volume of the tetrahedron is given by the absolute value of $D,$ where $D=\frac{1}{6}\left|\begin{array}{llll}x_{1} & y_{1} & z_{1} & 1 \\ x_{2} & y_{2} & z_{2} & 1 \\ x_{3} & y_{3} & z_{3} & 1 \\ x_{4} & y_{4} & z_{4} & 1\end{array}\right|$ Use this formula to find the volume of the tetrahedron with vertices (0,0,8),(2,8,0),(10,4,4) and (4,10,6)

A tetrahedron (triangular pyramid) has vertices $\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right),\left(x_{3}, y_{3}, z_{3}\right)$
and $\left(x_{4}, y_{4}, z_{4}\right)$. The volume of the tetrahedron is given by the absolute value of $D,$ where $D=\frac{1}{6}\left|\begin{array}{llll}x_{1} & y_{1} & z_{1} & 1 \\ x_{2} & y_{2} & z_{2} & 1 \\ x_{3} & y_{3} & z_{3} & 1 \\ x_{4} & y_{4} & z_{4} & 1\end{array}\right|$
Use this formula to find the volume of the tetrahedron with vertices (0,0,8),(2,8,0),(10,4,4) and (4,10,6)

Key Concepts

Tetrahedron

A tetrahedron is a polyhedron with four vertices and four triangular faces. It is one of the simplest types of three-dimensional solids, and understanding its properties is essential in both geometry and spatial analysis.

Volume Calculation Using Determinants

The volume of a tetrahedron can be found by constructing a 4x4 matrix from the coordinates of its vertices (with an appended 1 for each vertex) and taking one-sixth of the absolute value of its determinant. This method simplifies the volume calculation in three-dimensional space using linear algebra techniques.

Matrix Determinants

A matrix determinant is a scalar value that can be computed from the elements of a square matrix. This value is used to measure scaling effects and transformations in vector spaces. In the context of geometry, determinants help in calculating volumes and areas by providing a precise measure of how a transformation stretches or shrinks space.

Transcript

00:01 This problem is an application of evaluating a 4x4 determinant.

00:06 In this problem, you are asked to find the volume of a tetrahedron given the four vertices.

00:15 You are also given a formula that says that the volume is the absolute value of 1 .6th of the value of a 4x4 determinant formed from the vertices.

00:28 And i have this determinant written over here and you can see the setup of it with the vertices.

00:37 The first vertex is x1, y, 1, z1, and then a one on the end and so on through this.

00:45 So what we are going to do on this problem is use the vertices that you're given for the tetrahedron, set up the 4x4 determinant, evaluate that determinant, and then take one sixth of its absolute value, and that would give us the volume for the tetrahedron.

01:07 So let's start off, and let's set up the 4x4 determinant.

01:11 Now, as i work through the problem, i'm going to ignore the 1 -6 for a while and just evaluate the 4x4 determinant.

01:21 Then if we finished evaluating it, we will take into consideration the 1 -6th to find the final volume.

01:31 Okay, so setting up the determinant using the pattern that we were given over here on the right and the vertices that we were given on the left, i have 0 -0 -8 -1, 2801, 10 -4 -4 -1, and 4 -10 -6 -1.

02:01 So this is the determinant that we're going to start off and evaluate.

02:05 Now, keep in mind, it is a 4x4.

02:08 It's a rather good size determinant.

02:10 So we've got to pick a row to use our co -factors from.

02:15 And since the first row has a couple c -rows in it, i'm going to use this as my row.

02:24 It doesn't matter which row you pick, but this one you'll see in a little bit is going to be the easiest of the four rows.

02:31 So we start off with the c -row.

02:34 So since the zero holds the spot of the first row, first column, i will ignore the numbers on the first row in the first column and have a determinant of the remaining numbers, which will be 8, 0, 1, 4, 1, 10, 6, 1.

03:03 Okay, alternating our signs.

03:08 Next i will have minus 0.

03:14 Okay, now that 0 is the first row second column, so i'll block out those numbers.

03:20 And have 2, 0, 1, 10, 4, 1, and then 4, 1.

03:34 All right, my next number is 8, alternating signs again.

03:40 8 is on the first row of 3rd.

03:41 Column so i'll block out the numbers on the first row third column and have 2 8 1 10 4 1 and 410 1 okay alternate signs again i'm gonna have a minus 1 this one is the first row 4th column so i block out the numbers on the first row and the fourth column and i'll have 280 0 10 4 4 and then last is 4, 10, 6.

04:25 Okay, so now i have a series of 3x3 determinants we'll have to evaluate.

04:30 Now, the good thing about picking the row that i picked, because of the zeros on these first two, i can pretty well ignore those.

04:39 Because we're going to multiply by 0, i'm going to have 0.

04:43 All right, so we're going to evaluate this 3x3 determinant and multiply by 8.

04:48 So i'm going to work through that totally before i work with the last one.

04:53 So we're going to have eight times.

04:58 Okay, i'll come back to that eight.

05:01 Okay, i'm going to pick the first row on this one to work with.

05:08 So i'm going to have two times the determinant...