Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.3: Vectors
Problem 11E
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![### Volume of a Parallelepiped
To find the volume of the parallelepiped determined by the vectors \(\vec{a} = (3, 4, -1)\), \(\vec{b} = (0, 2, 4)\), and \(\vec{c} = (2, 4, 1)\), follow these steps:
1. **Vectors Definition:**
- \(\vec{a} = (3, 4, -1)\)
- \(\vec{b} = (0, 2, 4)\)
- \(\vec{c} = (2, 4, 1)\)
2. **Volume Calculation:**
To find the volume of the parallelepiped, compute the scalar triple product of the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).
The scalar triple product is calculated as:
\[
\text{Volume} = |\vec{a} \cdot (\vec{b} \times \vec{c})|
\]
3. **Determining \(\vec{b} \times \vec{c}\):**
To find \(\vec{b} \times \vec{c}\):
\[
\vec{b} \times \vec{c} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & 2 & 4 \\
2 & 4 & 1
\end{vmatrix}
\]
4. **Determinant Calculation:**
Calculate the determinant:
\[
\vec{b} \times \vec{c} = \mathbf{i}(2 \cdot 1 - 4 \cdot 4) - \mathbf{j}(0 \cdot 1 - 4 \cdot 2) + \mathbf{k}(0 \cdot 4 - 2 \cdot 2)
\]
\[
\vec{b} \times \vec{c} = \mathbf{i}(2 - 16) - \mathbf{j}(0 - 8) + \mathbf{k}(0 - 4)
\]
\[
\vec{b} \times \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55d3466e-6a31-4aff-972b-e33941523058%2Fda1846e8-00ec-44f5-aa9c-c4120366d68e%2Fu0chd2y_processed.png&w=3840&q=75)
Transcribed Image Text:### Volume of a Parallelepiped
To find the volume of the parallelepiped determined by the vectors \(\vec{a} = (3, 4, -1)\), \(\vec{b} = (0, 2, 4)\), and \(\vec{c} = (2, 4, 1)\), follow these steps:
1. **Vectors Definition:**
- \(\vec{a} = (3, 4, -1)\)
- \(\vec{b} = (0, 2, 4)\)
- \(\vec{c} = (2, 4, 1)\)
2. **Volume Calculation:**
To find the volume of the parallelepiped, compute the scalar triple product of the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).
The scalar triple product is calculated as:
\[
\text{Volume} = |\vec{a} \cdot (\vec{b} \times \vec{c})|
\]
3. **Determining \(\vec{b} \times \vec{c}\):**
To find \(\vec{b} \times \vec{c}\):
\[
\vec{b} \times \vec{c} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & 2 & 4 \\
2 & 4 & 1
\end{vmatrix}
\]
4. **Determinant Calculation:**
Calculate the determinant:
\[
\vec{b} \times \vec{c} = \mathbf{i}(2 \cdot 1 - 4 \cdot 4) - \mathbf{j}(0 \cdot 1 - 4 \cdot 2) + \mathbf{k}(0 \cdot 4 - 2 \cdot 2)
\]
\[
\vec{b} \times \vec{c} = \mathbf{i}(2 - 16) - \mathbf{j}(0 - 8) + \mathbf{k}(0 - 4)
\]
\[
\vec{b} \times \
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