(a) Let a, b € Rm be nonzero vectors. Prove that removing a's projection on the line of b from a leaves a vector perpendicular to b: bb™ br (arba) = 0 (0.0.1) Note: be aware of the dimension of each part to make sure you are canceling the correct part. (b) Let b₁,b2,..., b; be a list of nonzero mutually orthogonal vectors of Rm. Prove that removing each of a's projections to some b; from a leaves a vector perpendicular to each of Let b₁,b2, ..., bj : br (a - Σ i=1 T bibi -a) = 0, (for all k = 1,2, ..., j). b₂¹b₁ (0.0.2) This is almost trivial algebraically. Do you understand the geometric mean- ing of this, for example in R³(when j = 2)? Trying to draw it might help. Hint: You can discuss the relationship between k and i. Use (a) when they are the same, and use orthogonality when they are different.

(a) Let a, b € Rm be nonzero vectors. Prove that removing a's projection on the line of b from a leaves a vector perpendicular to b: bb™ br (arba) = 0 (0.0.1) Note: be aware of the dimension of each part to make sure you are canceling the correct part. (b) Let b₁,b2,..., b; be a list of nonzero mutually orthogonal vectors of Rm. Prove that removing each of a's projections to some b; from a leaves a vector perpendicular to each of Let b₁,b2, ..., bj : br (a - Σ i=1 T bibi -a) = 0, (for all k = 1,2, ..., j). b₂¹b₁ (0.0.2) This is almost trivial algebraically. Do you understand the geometric mean- ing of this, for example in R³(when j = 2)? Trying to draw it might help. Hint: You can discuss the relationship between k and i. Use (a) when they are the same, and use orthogonality when they are different.

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter7: Triangles

Section: Chapter Questions

Problem 1RP: We mentioned in Section 7.5 that our algebraic treatment of vectors could be attributed, in part, to...

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