Question

Write equations (2.12) out in detail and solve the three simultaneous equations (say by determinants) for${\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}$in terms of${\mathbf{x}}_{\mathbf{1}}\mathbf{\text{'}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{\text{'}}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}\mathbf{\text{'}}$to verify equations (2.13) . Use your results in Problem 4.

Short answer

The statement has been verified.

Step-by-step solution

Given Information

In matrix A, each element is the component of vectors.

Definition of Orthonormal triad.

The Orthonormal vector triad is the unit vectors in the positive direction of the three mutually perpendicular axes.

Show that Column vectors of A are orthonormal.

In matrix A, each element is the component of vectors.

The formula for the cartesian vectors states that $V_i\text{'}=\sum _{j=1}^3{a}_{ij}{V}_j$

In matrix form it is written as$V\text{'}=A^{\tau }V$

Since, A is an orthogonal matrix.

$$\begin{gathered} A^{\tau }{V}^{\text{'}}=A{A}^{T}V \\ {A}^{\tau }{V}^{\text{'}}=IV \\ {A}^{\tau }{V}^{\text{'}}=V \end{gathered}$$

The equation becomes as follows

$$\begin{gathered} V_i\text{'}=\sum _{j=1}^3{\left(A^{\tau }\right)}_{ij}{V^{\text{'}}}_j \\ {V}_i\text{'}=\sum _{j=1}^3{\left(a\right)}_{ij}{V^{\text{'}}}_j \end{gathered}$$

Hence The statement has been verified.