Question

Generalize Problem 6 to three dimensions; to n dimensions.

Short answer

$C$ is an orthogonal matrix.

Step-by-step solution

Given information

C is a matrix whose columns are the components $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ of two perpendicular vectors, each of unit length.

Orthogonal matrix

An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix with orthonormal vectors in the columns and rows.

Calculate matrix  C

To generalize matrix $C$, we need to represent n-dimensional matrix. Therefore, we will need $n$orthogonal vectors $r_n$each with $n$components.

A n-dimensional matrix can be mathematically presented as

$C=(r_1{r}_2{r}_2.......r_n),where{r}_i^T{r}_j={\delta }_{ij},and{\delta }_{ij}=\left\{\begin{array}{c}1\\ 0\end{array}\begin{array}{c}i=j\\ i\ne j\end{array}\right.$

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

$C^{T}C=\left(\begin{array}{c}{r}_1^T\\ r_2^T\\ ⋰\\ ..\\ r_n^T\end{array}\right)\left(r_1{r}_2{r}_2.........r_n\right)$

Therefore, ${\left(C^{T}C\right)}_{ij}=r_i^T{r}_j={\delta }_{ij}$, which shows that $C$ is an orthogonal matrix.