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Problem 5. Let u be a unit vector in R3 and let P: R³ → R³ be defined by P(v) = v – (v • u)u. 1. Show that Pis linear. 2. Show that PoP = P. 1 3. Show that for any vector v E R°, P(v) is perpendicular to u.

Problem 5. Let u be a unit vector in R3 and let P: R³ → R³ be defined by P(v) = v – (v • u)u. 1. Show that Pis linear. 2. Show that PoP = P. 1 3. Show that for any vector v E R°, P(v) is perpendicular to u.

Advanced Engineering Mathematics
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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Problem 5. Let u be a unit vector in R3 and let P: R³ → R³ be defined by P(v) = v – (v • u)u. 1. Show that P is linear. 2. Show that PoP= P. 1 3. Show that for any vector v e R³, P(v) is perpendicular to u. 4. Show that if P(v) = 0 then v is a scalar multiple of u. 5. True or false: any vector v E R³ can be expressed as a sum v = w + w' where w is a scalar multiple of u and w' is perpendicular to u. Does the mapping P relate to this? 6. Suppose vi, v2 are two nonzero vectors perpendicular to u and to each other. What are the values of P on u, v1, v2?
Transcribed Image Text:Problem 5. Let u be a unit vector in R3 and let P: R³ → R³ be defined by P(v) = v – (v • u)u. 1. Show that P is linear. 2. Show that PoP= P. 1 3. Show that for any vector v e R³, P(v) is perpendicular to u. 4. Show that if P(v) = 0 then v is a scalar multiple of u. 5. True or false: any vector v E R³ can be expressed as a sum v = w + w' where w is a scalar multiple of u and w' is perpendicular to u. Does the mapping P relate to this? 6. Suppose vi, v2 are two nonzero vectors perpendicular to u and to each other. What are the values of P on u, v1, v2?
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