1. Show that the set P is an affine set of dimension 2. To this end, express it as x(0) + span(x(1) , x(2)), where x (0) ∈ P, and x(1) , x(2) are linearly independent vectors. 2. Find the minimum Euclidean distance from 0 to the set P, and a point that achieves the minimum distance.

1. Show that the set P is an affine set of dimension 2. To this end, express it as x(0) + span(x(1) , x(2)), where x (0) ∈ P, and x(1) , x(2) are linearly independent vectors. 2. Find the minimum Euclidean distance from 0 to the set P, and a point that achieves the minimum distance.

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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Question

(Affine sets and projections) Consider the set in R³ defined by the equation

1. Show that the set P is an affine set of dimension 2. To this end, express it as x⁽⁰⁾ + span(x⁽¹⁾ , x⁽²⁾), where x (0) ∈ P, and x⁽¹⁾ , x⁽²⁾ are linearly independent vectors.

2. Find the minimum Euclidean distance from 0 to the set P, and a point that achieves the minimum distance.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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Step 1: Analysis and Introduction:

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Step 2: Prove that the set P is affine set.

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Step 3: Express the point on P as given.

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Step 4: Introduce Lagrange multipliers and find the partial derivatives.

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Step 5: Find the point of minima

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Step 6: Find the minimum value of the function:

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