Question

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)

Short answer

The volume of the tetrahedron is 186.67 cubic units.

Step-by-step solution

Understand the Given Formula

The volume of the tetrahedron is given by the formula\[ V = \frac{1}{6} \left| \begin{array}{cccc} x_{1} & y_{1} & z_{1} & 1 \ x_{2} & y_{2} & z_{2} & 1 \ x_{3} & y_{3} & z_{3} & 1 \ x_{4} & y_{4} & z_{4} & 1 \end{array} \right| \]This is the determinant of a 4x4 matrix that includes the coordinates of the vertices.

Substitute the Given Coordinates into the Matrix

The vertices given are (0,0,8), (2,8,0), (10,4,4), and (4,10,6). Substitute these into the matrix:\[ \left| \begin{array}{cccc} 0 & 0 & 8 & 1 \ 2 & 8 & 0 & 1 \ 10 & 4 & 4 & 1 \ 4 & 10 & 6 & 1 \end{array} \right| \]

Calculate the Determinant of the Matrix

Compute the determinant of the 4x4 matrix:\[ \text{Det} = \left| \begin{array}{cccc} 0 & 0 & 8 & 1 \ 2 & 8 & 0 & 1 \ 10 & 4 & 4 & 1 \ 4 & 10 & 6 & 1 \end{array} \right| \]Use the cofactor expansion along the first row for easier computation.

Simplify Using Cofactors

Expand along the first row (0, 0, 8, 1):\[ \text{Det} = 0 \cdot \left| \begin{array}{ccc} 8 & 0 & 1 \ 4 & 4 & 1 \ 10 & 6 & 1 \end{array} \right| - 0 \cdot \left| \begin{array}{ccc} 2 & 0 & 1 \ 10 & 4 & 1 \ 4 & 6 & 1 \end{array} \right| + 8 \cdot \left| \begin{array}{ccc} 2 & 8 & 1 \ 10 & 4 & 1 \ 4 & 10 & 1 \end{array} \right| - 1 \cdot \left| \begin{array}{ccc} 2 & 8 & 0 \ 10 & 4 & 4 \ 4 & 10 & 6 \end{array} \right| \]This simplifies to:\[ 8 \left| \begin{array}{ccc} 2 & 8 & 1 \ 10 & 4 & 1 \ 4 & 10 & 1 \end{array} \right| - 1 \left| \begin{array}{ccc} 2 & 8 & 0 \ 10 & 4 & 4 \end{array} \right| \]

Calculate 3x3 Determinants

First, compute the determinant of\[ \left| \begin{array}{ccc} 2 & 8 & 1 \ 10 & 4 & 1 \ 4 & 10 & 1 \end{array} \right| \].This is:\[ 2(4\cdot1 - 1\cdot10) - 8(10\cdot1 - 4\cdot4) + 1(10\cdot10 - 4\cdot4) = 2(-6) - 8(-6) + 96 = -12 + 48 + 96 = 132. \]Next, compute the determinant of\[ \left| \begin{array}{ccc} 2 & 8 & 0 \ 10 & 4 & 4 \end{array} \right| \].This is:\[ 2(4\cdot6 - 4\cdot10 = 2(-32) = -64. \]

Compute the Overall Determinant

Combine the results:\[ 8 \cdot 132 - 1 \cdot (-64) = 1056 + 64 = 1120 \]

Calculate the Volume

Finally, the volume is given by the formula\[ V = \frac{1}{6} \left| 1120 \right| = \frac{1120}{6} = 186.67 \text{ units}^3 \]

Key concepts

Determinants

Determinants are a fundamental concept in linear algebra and are widely used in various areas such as geometry and calculus.
The determinant of a matrix is a scalar value that is computed from its elements.
It is denoted as det(A) or |A| for a matrix A.
In simpler terms, the determinant helps us understand the 'scale' of transformations described by the matrix.
It can tell us, for example, if a transformation preserves area or volume, or if it flips orientation.

Matrix Operations

Matrix operations, which include addition, subtraction, multiplication, and finding determinants, are key to solving many problems in mathematics and physics.
When we talk about matrices in the context of geometry, we often deal with the determinant to understand volume, area, or other transformations.
In our exercise, we used matrix determinants to calculate the volume of a tetrahedron.
The specific operation involved forming a 4x4 matrix from the coordinates of the tetrahedron's vertices and then calculating its determinant.
This is done using cofactor expansion along a row or column, which breaks the large matrix into smaller 3x3 matrices whose determinants are easier to calculate.

3D Geometry

3D geometry deals with shapes and figures in three-dimensional space.
A fundamental element of 3D geometry is the tetrahedron, a polyhedron with four triangular faces, six edges, and four vertices.
Understanding the precise location of these vertices in 3-dimensional space is essential for applications in fields like computer graphics, engineering, and more.
In our example, the tetrahedron's vertices are given in the form (x, y, z), representing coordinates in the 3-dimensional Cartesian coordinate system.
To find its volume, we apply principles from linear algebra, specifically using matrices and their determinants.

Volume Calculation

The volume of a tetrahedron can be calculated using the determinant of a 4x4 matrix formed from its vertices.
The general formula is
\[ V = \frac{1}{6} \left| \begin{array}{cccc} x_{1} & y_{1} & z_{1} & 1 \ x_{2} & y_{2} & z_{2} & 1 \ x_{3} & y_{3} & z_{3} & 1 \ x_{4} & y_{4} & z_{4} & 1 \end{array} \right| \]
This determinant measures the 'signed volume' of the tetrahedron, and taking its absolute value ensures we get a positive volume.
The division by 6 comes from a geometric property of tetrahedrons, as they can be seen as a simpler fraction of a parallelepiped (a type of prism).
In our solved example, after forming the matrix and calculating its determinant using cofactor expansion, we got the final volume through this formula.