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Exercise 6. Show that the vectors uj = (1,2, 3), uz = (-1,–1,1) in R³ are perpendicular. Find a nonzero vector uz in R' perpendicular to both. Express the vector w = (1,1,1) as a linear combination of u1, u2, uz. (Does it help to think of uj as i+2j + 3k and uz as -i – j + k here?) %3D

Exercise 6. Show that the vectors uj = (1,2, 3), uz = (-1,–1,1) in R³ are perpendicular. Find a nonzero vector uz in R' perpendicular to both. Express the vector w = (1,1,1) as a linear combination of u1, u2, uz. (Does it help to think of uj as i+2j + 3k and uz as -i – j + k here?) %3D

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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solve exercies 6 with explanation asap
The V in exercises is a notation used to denote the subset of vectors in Hamilton's quaternions. We didn't see matrices, so please utilize the Hamilton's approach. Thanks Exercise 3. What quaternions commute with all vectors v e V? Exercise 4. Is it possible to have nonzero vectors u, v in V with uv = 0? Why or why not? (Here the ordinary quaternion product is meant.) Exercise 5. Suppose v is a nonzero vector. Show that any vector w can be written as w = av+u where u is perpendicular to v. How can you express a in terms of v and w? (Hint: is there a scalar a so w – av is perpendicular to v? What equation does answering that require solving? Try some examples if this hint doesn't cause things to "click".) (Note that this would be more cumbersome to state if we didn't admit the zero vector to be perpendicular to every vector.) Exercise 6. Show that the vectors u = (1,2, 3), u2 = (-1,–1,1) in R³ are perpendicular. Find a nonzero vector uz in R perpendicular to both. Express the vector w = (1,1, 1) as a linear combination of u1, u2, uz. (Does it help to think of u as i+2j + 3k and uz as -i– j+k here?) Exercise 7. It's easy to solve the problem: express a given vector w as a linear combination of perpendicular unit vectors u and v. How does that go again? Now can this idea be extended and/or modified to deal with any of the following? 1. More than two perpendicular unit vectors 2. Two perpendicular vectors u and v which may not be unit vectors 3. Expressing w as a linear combination of some number l of perpendicular vectors u1, u2, ..., u Finally, I didn't specify whether I meant the vectors to be in R² or R³ (or C or V). Does it make a difference in the answer?
Transcribed Image Text:The V in exercises is a notation used to denote the subset of vectors in Hamilton's quaternions. We didn't see matrices, so please utilize the Hamilton's approach. Thanks Exercise 3. What quaternions commute with all vectors v e V? Exercise 4. Is it possible to have nonzero vectors u, v in V with uv = 0? Why or why not? (Here the ordinary quaternion product is meant.) Exercise 5. Suppose v is a nonzero vector. Show that any vector w can be written as w = av+u where u is perpendicular to v. How can you express a in terms of v and w? (Hint: is there a scalar a so w – av is perpendicular to v? What equation does answering that require solving? Try some examples if this hint doesn't cause things to "click".) (Note that this would be more cumbersome to state if we didn't admit the zero vector to be perpendicular to every vector.) Exercise 6. Show that the vectors u = (1,2, 3), u2 = (-1,–1,1) in R³ are perpendicular. Find a nonzero vector uz in R perpendicular to both. Express the vector w = (1,1, 1) as a linear combination of u1, u2, uz. (Does it help to think of u as i+2j + 3k and uz as -i– j+k here?) Exercise 7. It's easy to solve the problem: express a given vector w as a linear combination of perpendicular unit vectors u and v. How does that go again? Now can this idea be extended and/or modified to deal with any of the following? 1. More than two perpendicular unit vectors 2. Two perpendicular vectors u and v which may not be unit vectors 3. Expressing w as a linear combination of some number l of perpendicular vectors u1, u2, ..., u Finally, I didn't specify whether I meant the vectors to be in R² or R³ (or C or V). Does it make a difference in the answer?
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