In this exercise you will explore the ideas of orthogonal vectors, orthogonal matrices, and why these ideas are useful:(a) Plot the vectors v₁ =(1)in the xy-plane.1(b) Compute v₁ V₁ and v₂ v₂. The fact that you get an answer of 1 for both is due to the length of the vectors beingequal to 1. Now compute v₁ · v₂ and v2 - V₁. The fact that you get an answer of 0 for both of these answers is due tothe fact that these two vectors are perpendicular or "orthogonal" to one another.(V1 V₂)(1whose columns are v₁ and v₁ respectively and the matrix, V2 =(c) Now consider the matrix A==√/2=V1ᎪᎢ√2V₂A AT and AT A. What does this tell you in terms of inverse matrices?(d) What do you think is the general fact one can say about how to construct these nice “orthogonal matrices" like A ingeneral? How would you go about it (in words, don't actually come up with an example) for a 3 × 3 matrix?whose rows are v₁ and v₂ transposed (i.e. turned sideways) respectively. Compute

Answered: In this exercise you will explore the…

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

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In this exercise you will explore the ideas of orthogonal vectors, orthogonal matrices, and why these ideas are useful:
(a) Plot the vectors v₁ =
(1)
in the xy-plane.
1
(b) Compute v₁ V₁ and v₂ v₂. The fact that you get an answer of 1 for both is due to the length of the vectors being
equal to 1. Now compute v₁ · v₂ and v2 - V₁. The fact that you get an answer of 0 for both of these answers is due to
the fact that these two vectors are perpendicular or "orthogonal" to one another.
(V1 V₂)
(1
whose columns are v₁ and v₁ respectively and the matrix
, V2 =
(c) Now consider the matrix A
=
=
√/2
=
V1
ᎪᎢ
√2
V₂
A AT and AT A. What does this tell you in terms of inverse matrices?
(d) What do you think is the general fact one can say about how to construct these nice “orthogonal matrices" like A in
general? How would you go about it (in words, don't actually come up with an example) for a 3 × 3 matrix?
whose rows are v₁ and v₂ transposed (i.e. turned sideways) respectively. Compute

Expert Solution

Introduction

note :

Since you have posted question with multiple sub parts, we will provide the solution only to the first three sub parts of the question as per Q&A guidelines. Please repost the remaining question by specifying the sub parts need to be answered.

In this problem, we will be working with two vectors v₁ and v₂ in the xy-plane, as well as a matrix A formed from these two vectors.

We will first plot the vectors in the xy-plane, then compute the dot product of each vector with itself and with each other. Finally, we will compute the dot product of A with its transpose and the dot product of the transpose of A with A to understand the properties of these matrices.