20. Find the volume of the parallelepiped with adjacent edges u = (-2, 0.75, 4), v = (6, -0.3, 8), and w = (3, -2.5, 9).
20. Find the volume of the parallelepiped with adjacent edges u = (-2, 0.75, 4), v = (6, -0.3, 8), and w = (3, -2.5, 9).
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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![**Mathematics Practice Problems**
**Problem 18: Finding Surface Area of a Parallelepiped**
Find the surface area of the parallelepiped with adjacent edges given by:
\[ \mathbf{u} = \langle 4, 7, -8 \rangle, \ \mathbf{v} = \langle -2, 5, 11 \rangle, \ \mathbf{w} = \langle 9, -2, -8 \rangle \]
**Solution Process:**
1. Calculate the area of each pair of adjacent edges by taking the magnitude of their cross product.
2. Sum the areas of the faces.
**Problem 19: Distance Between Two Points in 3D Space**
The position of one airplane is represented by the coordinates \((11, 10, 3)\), and a second airplane is represented by the coordinates \((-9, 14, 3)\). Determine the distance between the planes if one unit represents one mile.
**Solution Process:**
1. Use the distance formula for 3D points: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
**Problem 20: Finding Volume of a Parallelepiped**
Find the volume of the parallelepiped with adjacent edges given by:
\[ \mathbf{u} = \langle -2, 0.75, 4 \rangle, \ \mathbf{v} = \langle 6, -0.3, 8 \rangle, \ \mathbf{w} = \langle 3, -2.5, 9 \rangle \]
**Solution Process:**
1. Calculate the scalar triple product of the vectors by computing the determinant of the matrix formed by the vectors.
2. The absolute value of this determinant gives the volume.
These practice problems cover the applications of vectors, distances in 3D space, and determinants for computing volumes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b3799ad-3ea5-4cad-a42a-63419fcded6b%2F6415c9c5-325f-4a20-99fe-4153548a146a%2F5ltybie_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics Practice Problems**
**Problem 18: Finding Surface Area of a Parallelepiped**
Find the surface area of the parallelepiped with adjacent edges given by:
\[ \mathbf{u} = \langle 4, 7, -8 \rangle, \ \mathbf{v} = \langle -2, 5, 11 \rangle, \ \mathbf{w} = \langle 9, -2, -8 \rangle \]
**Solution Process:**
1. Calculate the area of each pair of adjacent edges by taking the magnitude of their cross product.
2. Sum the areas of the faces.
**Problem 19: Distance Between Two Points in 3D Space**
The position of one airplane is represented by the coordinates \((11, 10, 3)\), and a second airplane is represented by the coordinates \((-9, 14, 3)\). Determine the distance between the planes if one unit represents one mile.
**Solution Process:**
1. Use the distance formula for 3D points: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
**Problem 20: Finding Volume of a Parallelepiped**
Find the volume of the parallelepiped with adjacent edges given by:
\[ \mathbf{u} = \langle -2, 0.75, 4 \rangle, \ \mathbf{v} = \langle 6, -0.3, 8 \rangle, \ \mathbf{w} = \langle 3, -2.5, 9 \rangle \]
**Solution Process:**
1. Calculate the scalar triple product of the vectors by computing the determinant of the matrix formed by the vectors.
2. The absolute value of this determinant gives the volume.
These practice problems cover the applications of vectors, distances in 3D space, and determinants for computing volumes.
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