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 theHamilton's approach. ThanksExercise 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? (Herethe ordinary quaternion product is meant.)Exercise 5. Suppose v is a nonzero vector. Show that any vector w can be written asw = av+uwhere u is perpendicular to v. How can you express a in terms of v and w? (Hint: is there a scalara so w – av is perpendicular to v? What equation does answering that require solving? Try someexamples 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 beperpendicular to every vector.)Exercise 6. Show that the vectors u = (1,2, 3), u2 = (-1,–1,1) in R³ are perpendicular. Finda nonzero vector uz in R perpendicular to both. Express the vector w = (1,1, 1) as a linearcombination 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 ofperpendicular unit vectors u and v. How does that go again? Now can this idea be extended and/ormodified to deal with any of the following?1. More than two perpendicular unit vectors2. Two perpendicular vectors u and v which may not be unit vectors3. Expressing w as a linear combination of some number l of perpendicular vectors u1, u2, ..., uFinally, I didn't specify whether I meant the vectors to be in R² or R³ (or C or V). Does it make adifference in the answer?

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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 theHamilton's approach. ThanksExercise 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? (Herethe ordinary quaternion product is meant.)Exercise 5. Suppose v is a nonzero vector. Show that any vector w can be written asw = av+uwhere u is perpendicular to v. How can you express a in terms of v and w? (Hint: is there a scalara so w – av is perpendicular to v? What equation does answering that require solving? Try someexamples if this hint doesn't cause things to &#34;click&#34;.)(Note that this would be more cumbersome to state if we didn't admit the zero vector to beperpendicular to every vector.)Exercise 6. Show that the vectors u = (1,2, 3), u2 = (-1,–1,1) in R³ are perpendicular. Finda nonzero vector uz in R perpendicular to both. Express the vector w = (1,1, 1) as a linearcombination 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 ofperpendicular unit vectors u and v. How does that go again? Now can this idea be extended and/ormodified to deal with any of the following?1. More than two perpendicular unit vectors2. Two perpendicular vectors u and v which may not be unit vectors3. Expressing w as a linear combination of some number l of perpendicular vectors u1, u2, ..., uFinally, I didn't specify whether I meant the vectors to be in R² or R³ (or C or V). Does it make adifference in the answer?

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Answered: Image /qna-images/answer/48aa3ff2-b091-485d-8878-7889bfafa753.jpg

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