Question

Any rotation of axes in three dimensions can be described by giving the nine direction cosines of the angle between the $\left(x,y,z\right)$axes and the $\left(x\text{'},y\text{'},z\text{'}\right)$axes. Show that the matrix A of these direction cosines in $\left(2.7\right)$or $\left(2.10\right)$is an orthogonal matrix. Hint: See Chapter 3, Section 9. Find $\mathit{A}{\mathit{A}}^{\mathbf{T}}$and use Problem 3.

Short answer

Answer

The statement is verified and$A{A}^T=I$

Step-by-step solution

Given Information

Matrix A with direction cosines$\left(x,y,z\right),\left(x\text{'},y\text{'},z\text{'}\right)$.

Definition of Orthogonal matrix.

A Square matrix is said to be orthogonal matrix if $\mathit{A}{\mathit{A}}^{\mathbf{T}}\mathbf{=}\mathit{I}$.

Show that A is orthogonal.

Matrix A with direction cosines $\left(x,y,z\right),\left(x\text{'},y\text{'},z\right)$.

$$\begin{gathered} A-\left(a_{,},a_2,a_3\right) \\ {A}^T=\left(\begin{array}{c}{a}_1^T\\ a_2^T\\ a_3^T\end{array}\right) \end{gathered}$$

The value of $A{A}^T$is as follows.

$$\begin{gathered} A{A}^T=\underset{k=1}{\overset{3}{\sum a_{ki}{a}_{kj}}} \\ A{A}^I=a_i^I{a}_j \\ A{A}^T={\delta }_{ij} \end{gathered}$$

Hence, the statement is verified and $A{A}^T=I$.