Find the volume of the parallelepiped (box) determined by u, v, and w. u v w i+ 2j 2i - 2j + 3k 2i + k The volume of the parallelepiped is (Simplify your answer.) units cubed.
Find the volume of the parallelepiped (box) determined by u, v, and w. u v w i+ 2j 2i - 2j + 3k 2i + k The volume of the parallelepiped is (Simplify your answer.) units cubed.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Finding the Volume of a Parallelepiped**
To find the volume of the parallelepiped (box) determined by the vectors **u**, **v**, and **w** given in the question, follow these steps:
Vectors:
- **u** = i + 2j
- **v** = 2i - 2j + 3k
- **w** = 2i + k
The volume of the parallelepiped is determined by the scalar triple product of the vectors **u**, **v**, and **w**. The scalar triple product can be found using the determinant of a 3x3 matrix composed of the components of the vectors.
\[
\text{Volume} = \left| \begin{array}{ccc}
1 & 2 & 0 \\
2 & -2 & 3 \\
2 & 0 & 1 \\
\end{array} \right|
\]
The determinant of this matrix is computed as follows:
1. Multiply the diagonals from the top left to the bottom right and sum them up:
(1 * (-2) * 1) + (2 * 3 * 2) + (0 * 2 * 2) = -2 + 12 + 0 = 10
2. Multiply the diagonals from the top right to the bottom left and sum them up:
(0 * (-2) * 2) + (2 * 3 * 1) + (1 * 2 * 0) = 0 - 12 + 0 = -12
3. Subtract the sum of the second set of products from the first:
|10 - (-12)| = 22
Thus, the volume of the parallelepiped is:
\[
\boxed{22} \text{ units cubed}
\]
For further comprehension, you can refer to a graph or visualization of the vectors in 3D space, where you can see how the vectors form the edges of a parallelepiped and how the volume calculation is derived from the determinant of their components.
For additional practice with vector products and determinants, refer to the provided supplementary materials or practice exercises.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56c39839-d6ec-4cd3-94f9-6bb9a1cb44c1%2F6b52be4f-5773-43c5-bfcf-b91a58a1ac87%2Fd5ycqx5_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Volume of a Parallelepiped**
To find the volume of the parallelepiped (box) determined by the vectors **u**, **v**, and **w** given in the question, follow these steps:
Vectors:
- **u** = i + 2j
- **v** = 2i - 2j + 3k
- **w** = 2i + k
The volume of the parallelepiped is determined by the scalar triple product of the vectors **u**, **v**, and **w**. The scalar triple product can be found using the determinant of a 3x3 matrix composed of the components of the vectors.
\[
\text{Volume} = \left| \begin{array}{ccc}
1 & 2 & 0 \\
2 & -2 & 3 \\
2 & 0 & 1 \\
\end{array} \right|
\]
The determinant of this matrix is computed as follows:
1. Multiply the diagonals from the top left to the bottom right and sum them up:
(1 * (-2) * 1) + (2 * 3 * 2) + (0 * 2 * 2) = -2 + 12 + 0 = 10
2. Multiply the diagonals from the top right to the bottom left and sum them up:
(0 * (-2) * 2) + (2 * 3 * 1) + (1 * 2 * 0) = 0 - 12 + 0 = -12
3. Subtract the sum of the second set of products from the first:
|10 - (-12)| = 22
Thus, the volume of the parallelepiped is:
\[
\boxed{22} \text{ units cubed}
\]
For further comprehension, you can refer to a graph or visualization of the vectors in 3D space, where you can see how the vectors form the edges of a parallelepiped and how the volume calculation is derived from the determinant of their components.
For additional practice with vector products and determinants, refer to the provided supplementary materials or practice exercises.
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