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.

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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

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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

Fields in Physics

Engineering Physics

Astrophysics

Electricity and Magnetism

Electrostatics

Torque and Rotational Motion

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 $C^{T} C$