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Chapter 15: Problem 39

Find the volume of the tetrahedron shown in the figure. Its corners are \((0,0,0),(1,0,0),(0,1,0)\), and \((0,0,1)\).

Short Answer

Expert verified
The volume of the tetrahedron with vertices at (0,0,0), (1,0,0), (0,1,0), and (0,0,1) is \(\dfrac{1}{3}\) cubic units.

Step by step solution

01

Creating Position Vectors

Write the position vectors \(\textbf{a}\), \(\textbf{b}\), and \(\textbf{c}\) for the given vertices (1,0,0), (0,1,0), and (0,0,1). \(\textbf{a} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\) \(\textbf{b} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) \(\textbf{c} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\)
02

Evaluating Cross Product

Find the cross product of the vectors \(\textbf{b}\) and \(\textbf{c}\), denoted by \(\textbf{b} \times \textbf{c}\). \(\textbf{b} \times \textbf{c} = \begin{pmatrix} i & j & k \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\)
03

Evaluating Dot Product

Find the dot product of vector \(\textbf{a}\) with the cross product from Step 2, denoted by \(\textbf{a} \cdot (\textbf{b} \times \textbf{c})\). \(\textbf{a} \cdot (\textbf{b} \times \textbf{c}) = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = 1\)
04

Applying the Volume Formula

Use the volume formula and the result from Step 3 to find the final volume of the tetrahedron. Volume = \(\dfrac{1}{3}|\textbf{a} \cdot (\textbf{b} \times \textbf{c})|\) Volume = \(\dfrac{1}{3}|1|\) Volume = \(\dfrac{1}{3}\) cubic units The volume of the tetrahedron is \(\dfrac{1}{3}\) cubic units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Position Vectors
In geometry, position vectors are crucial for identifying points in space. They are vectors that carry the coordinates of a point relative to a specific origin. When you have a tetrahedron with vertices at \(0, 0, 0\), \(1, 0, 0\), \(0, 1, 0\), and \(0, 0, 1\), each vertex can be expressed as a position vector.
For example:
  • \[\textbf{a} = \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}\] represents the vector from the origin to the vertex \(1, 0, 0\).
  • \[\textbf{b} = \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}\] indicates the point \(0, 1, 0\).
  • \[\textbf{c} = \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}\] denotes the position \(0, 0, 1\).
These vectors allow us to perform operations like cross and dot products, essential steps in computing volume.
Cross Product
The cross product is a vital operation in vector algebra, used to find a vector perpendicular to two given vectors. When finding the volume of a tetrahedron, the cross product helps create a base area oriented in 3D space.
Given vectors \(\textbf{b}\) and \(\textbf{c}\), the calculation of \(\textbf{b} \times \textbf{c}\) is performed by constructing a determinant:
  • List the unit vectors \(i\), \(j\), and \(k\) as the first row.
  • Assign \(\textbf{b} = \begin{pmatrix} 0 & 1 & 0 \end{pmatrix}\) as the second row and \(\textbf{c} = \begin{pmatrix} 0 & 0 & 1 \end{pmatrix}\) as the third row.
Upon resolving, \(\textbf{b} \times \textbf{c}\) results in \(\begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}\). This operation provides the direction and magnitude needed for determining volume.
Dot Product
The dot product is another key operation in vector mathematics, used to project one vector onto another. It measures how much a vector \(\textbf{a}\) lies in the direction of \(\textbf{b} \times \textbf{c}\), particularly when finding volumes.
To calculate this, align the vector \(\textbf{a} = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix}\) with the resulting cross product \(\textbf{b} \times \textbf{c} = \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}\):
  • Multiply corresponding components: \(1\cdot1 + 0\cdot0 + 0\cdot0 = 1\).
  • Sum the products to get the scalar value.
This dot product determines how the vectors stack to enclose space, integral to finding the volume of geometric shapes.
Volume Formula
Calculating the volume of a tetrahedron might seem complex, yet it becomes accessible using vectors. The formula requires position vectors, cross products, and dot products:
  • Start with the scalar from the dot product result, \(1\).
  • Apply the volume formula: \[\text{Volume} = \dfrac{1}{3}|\textbf{a} \cdot (\textbf{b} \times \textbf{c})|\]
  • Substitute the calculated dot product into the formula: \[\text{Volume} = \dfrac{1}{3}|1|=\dfrac{1}{3} \text{ cubic units}\]
This formula takes advantage of vector operations to compute the exact volume efficiently, avoiding complex geometric or calculus problems.

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