91Ó°ÊÓ

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.

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

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.

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

Hence, the equation has been verified.