Question

Verify equations(2.6).

Short answer

The equation has been verified.

Step-by-step solution

Given Information

The unit basis vectors are $i,j,k$along $(x,y,z)$axis and the unit basis vectors are $i^{\text{'}},j^{\text{'}},k^{\text{'}}$along $(x\text{'},y\text{'},z\text{'})$axis.

Definition of Cartesian vector.

Under the rotation of rectangular axes, the vectors transform in the same way as displacement vectors, which is called cartesian vectors.

Verify the equation.

The unit basis vectors are $i,j,k$along $(x,y,z)$axis and the unit basis vectors are $i\text{'},j\text{'},k\text{'}$along $(x\text{'},y\text{'},z\text{'})$axis.

Let represents the coordinates system.

$\begin{array}{l}r=xi+yj+zk\\ r=x^{\text{'}}{i}^{\text{'}}+y^{\text{'}}{j}^{\text{'}}{z}^{\text{'}}{k}^{\text{'}}\end{array}$

Dot the above equation to the $i,j,k$respectively.

The equation becomes as follows.

$\begin{array}{l}x=x\text{'}i{i}^{\text{'}}+y\text{'}ij+z\text{'}ik\\ y=x^{\text{'}}j{i}^i+y\text{'}j{j}^{\text{'}}+z\text{'}{k}^{\text{'}}\\ z=x^{\text{'}}k{i}^{\text{'}}+y^{\text{'}}k{j}^{\text{'}}+z^{\text{'}}k{k}^{\text{'}}\end{array}$

Substitute Values mentioned below in the above equation.

$\begin{array}{l}i{i}^{\text{'}}=l_1\\ i{j}^{\text{'}}=I_2\\ i{k}^{\text{'}}=I_3\\ j{i}^{\text{'}}=m_1\end{array}$

Simplify Further

$\begin{array}{l}j{j}^{\text{'}}=m_2\\ i{k}^{\text{'}}=m_3\\ k{i}^{\text{'}}=n_2\\ k{j}^{\text{'}}=n_2\end{array}$

The Value is $kk\text{'}=k_3$

The equation becomes as follows.

$\begin{array}{l}x=x^{\text{'}}{l}_1+y^{\text{'}},i{l}_2+z\text{'}{l}_3\\ y=x\text{'}{m}_1+y\text{'}{m}_2+z\text{'}{m}_3\\ z=x^{\text{'}}{n}_{1\partial }+y\text{'}{n}_2+z\text{'}{n}_3\end{array}$

Hence, the equation has been verified.