Consider a parallelogram containing vertices at the following points: (2, 1,3), (3, 4, 5), and (0,q,0). (a) Find the value of q that makes a parallelogram with minimum area. (b) Using the value of q from the previous part, determine the minimum area.
Consider a parallelogram containing vertices at the following points: (2, 1,3), (3, 4, 5), and (0,q,0). (a) Find the value of q that makes a parallelogram with minimum area. (b) Using the value of q from the previous part, determine the minimum area.
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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Transcribed Image Text:### Problem Statement
Consider a parallelogram containing vertices at the following points: \((2, 1, 3)\), \((3, 4, 5)\), and \((0, q, 0)\).
**(a)** Find the value of \( q \) that makes a parallelogram with **minimum** area.
**(b)** Using the value of \( q \) from the previous part, **determine the minimum area**.
### Solution
*(To guide students through the problem, a step-by-step solution can be provided, including mathematical derivations and formulae related to finding the area of a parallelogram in 3D space and optimization techniques. Visual aids like diagrams of the parallelogram demonstrating how different values of \( q \) affect the area can also be included.)*
For now, though, the problem requires identifying and solving for the specific value of \( q \), ensuring students use the cross product to find the area of the parallelogram and processes such as differentiation to find the minimum possible value.
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