Q6P Show that if 聽C聽is a matrix wh... [FREE SOLUTION]

Chapter 1: Q6P (page 1)

Show that if $C$ is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors each of unit length, then $C$ is an orthogonal matrix. Hint: Find $C^{T} C$

Short Answer

Expert verified

$C$ is an orthogonal matrix.

Step by step solution

Given information

C is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors, each of unit length.

Orthogonal matrix

An orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

Calculate the matrix C

Write matrix $C$ as $C = (r_{1}, r_{2})$ , where $r_{1}$ and $r_{2}$ are orthogonal unit vectors, that is,

${r_{1}}^{T} r_{2} = 0 {r_{1}}^{T} r_{1} = 1 {r_{2}}^{T} r_{2} = 1$

Now, calculate $C^{T} C$ .

$C^{T} C = (\begin{matrix} r_{1}^{T} \\ r_{2}^{2} \end{matrix}) (\begin{matrix} r_{1} & r_{2} \end{matrix}) = (\begin{matrix} r_{1}^{T} r_{1} \\ r_{2}^{T} r_{1} \end{matrix} \begin{matrix} r_{1}^{T} r_{2} \\ r_{1}^{T} r_{1} \end{matrix})$

Use the relations above for the orthonormality of $r_{1}$ and $r_{2}$ .

$C^{T} C = (\begin{matrix} 1 & 0 \\ n & 1 \end{matrix})$

The above equation shows that $C$ is an orthogonal matrix.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Show that if $C$ is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors each of unit length, then $C$ is an orthogonal matrix. Hint: Find $C^{T} C$

Short Answer

Step by step solution

Given information

Orthogonal matrix

Calculate the matrix C

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Rotational Dynamics

Geometrical and Physical Optics

Nuclear Physics

Physics of Motion

Fluids

Study anywhere. Anytime. Across all devices.

91影视

Show that if Cis a matrix whose columns are the components (x1,y1)and (x2,y2)of two perpendicular vectors each of unit length, then Cis an orthogonal matrix. Hint: FindCTC

Short Answer

Step by step solution

Given information

Orthogonal matrix

Calculate the matrix C

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Rotational Dynamics

Geometrical and Physical Optics

Nuclear Physics

Physics of Motion

Fluids

Study anywhere. Anytime. Across all devices.

Show that if $C$ is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors each of unit length, then $C$ is an orthogonal matrix. Hint: Find $C^{T} C$