Verify equations (7.13)using Figure 7.5. Hints: Write ${\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{r}$ as ${\mathbf{i}}^{\mathbf{\text{'}}}{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{+}{\mathbf{j}}^{\mathbf{\text{'}}}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{ix}\mathbf{+}\mathbf{jy}$ and take the dot product of this equation with${\mathbf{i}}^{\mathbf{\text{'}}}$and with${\mathbf{j}}^{\mathbf{\text{'}}}$to get${\mathbf{x}}^{\mathbf{\text{'}}}$and ${\mathbf{y}}^{\mathbf{\text{'}}}$ . Evaluate the dot products of the unit vectors in terms of$\mathbf{\theta }$using Figure 7.5. For example, ${\mathbf{i}}^{\mathbf{\text{'}}}\mathbf{\times }\mathbf{j}$ is the cosine of the angle between the x' axis and the yaxis.

The dot product of two vectors $\overrightarrow{\mathbf{u}}$and $\overrightarrow{\mathbf{v}}$, written as $\overrightarrow{\mathbf{u}}\mathbf{.}\overrightarrow{\mathbf{v}}$is defined by $\overrightarrow{\mathbf{u}}\mathbf{.}\overrightarrow{\mathbf{v}}\mathbf{=}\left|\overrightarrow{u}\right|\left|\overrightarrow{v}\right|$, where $\mathbf{\theta }$is the angle between $\overrightarrow{\mathbf{u}}$and $\overrightarrow{\mathbf{v}}$.

The vector r = ( x,y) and the vector ${\mathrm{r}}^{\text{'}}=({\mathrm{x}}^{\text{'}},{\mathrm{y}}^{\text{'}})$ are the same vectors, but their components are relative to different axes.

As the vector r = (x,y) and the vector $r^{\text{'}}=\left(x^{\text{'}}{y}^{\text{'}}\right)$are the same vectors. We write

$$\begin{gathered} \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{r}} \\ {\mathrm{i}}^{\text{'}}{\mathrm{x}}^{\text{'}}+{\mathrm{j}}^{\text{'}}{\mathrm{y}}^{\text{'}}=\mathrm{ix}+\mathrm{jy} \end{gathered}$$

Apply the dot product with ${\mathrm{i}}^{\text{'}}$, and it becomes

$\left({\mathrm{i}}^{\text{'}}{\mathrm{i}}^{\text{'}}\right){\mathrm{x}}^{\text{'}}+({\mathrm{J}}^{\text{'}}.{\mathrm{i}}^{\text{'}})=\left(\mathrm{i}.{\mathrm{i}}^{\text{'}}\right){\mathrm{x}}^{\text{'}}+(\mathrm{j}.{\mathrm{i}}^{\text{'}})\mathrm{y}$ (1)

From the figure (7.5) , the angle between the axes x and x' of the two unit vectors and i' is $\mathrm{\theta }$, so that

The dot product of the unit vectors i and i' is $=\left|\mathrm{i}\right|\left|\mathrm{i}\text{'}\right|\mathrm{cos\theta }=\mathrm{cos\theta }$, where $\left|\mathrm{i}\right|=1$and $\left|\mathrm{i}\text{'}\right|=1$

And the angle between the axes y and x' of the two unit vectors j and i' is $\left(\frac{\prod _{}}{2}-\mathrm{\theta }\right)=\mathrm{sin\theta }$, so that the dot product of the unit vectors and is , where $\left|\mathrm{j}\right|=1,\left|\mathrm{i}\text{'}\right|=1$and we use the following relation

$$\begin{gathered} \cos \left(\frac{\prod _{}}{2}-0\right)=\cos \left(\frac{\prod _{}}{2}\right)\times \mathrm{cos\theta }+\sin \left(\frac{\prod _{}}{2}\right)\times \mathrm{sin\theta } \\ =0\times \mathrm{cos\theta }+\mathrm{sin\theta } \\ =\mathrm{sin\theta } \end{gathered}$$

Also, the angle between the axes and of the two unit vectors j' and i' is $\frac{\prod _{}}{2}$, so that the dot product of the unit vectors and is $\mathrm{j}\text{'}.\mathrm{i}\text{'}=\left|\mathrm{j}\text{'}\right|\left|\mathrm{i}\text{'}\right|\cos \left(\frac{\prod _{}}{2}\right)=0$.

Finally, the dot product of the unit vectors and must be $\mathrm{i}\text{'}.\mathrm{i}\text{'}=\left|\mathrm{i}\text{'}\right|\left|\mathrm{i}\text{'}\right|\cos 0=1$, where $\left|\mathrm{i}\text{'}\right|=1$

Using the following equation in the equation (1) implies that

As the vector $\mathrm{r}=\left(\mathrm{x},\mathrm{y}\right)$and the vector $\mathrm{r}\text{'}=\left(\mathrm{x}\text{'},\mathrm{y}\text{'}\right)$are the same vectors. We write $$\begin{gathered} \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{r}} \\ \mathrm{i}\text{'}\mathrm{x}\text{'}+\mathrm{j}\text{'}\mathrm{y}\text{'}=\mathrm{ix}+\mathrm{jy} \end{gathered}$$

Apply the dot product with j', and it becomes

$\left(\mathrm{i}\text{'}.\mathrm{j}\text{'}\right)\mathrm{x}\text{'}+\left(\mathrm{j}\text{'}.\mathrm{j}\text{'}\right)\mathrm{y}\text{'}=\left(\mathrm{i}\text{'}.\mathrm{j}\text{'}\right)\mathrm{x}\text{'}+\left(\mathrm{j}\text{'}.\mathrm{j}\text{'}\right)\mathrm{y}\text{'}\left(2\right)$

From the figure , the angle between the axes $x^{\text{'}}$and $y^{\text{'}}$ of the two unit vectors $i^{\text{'}}$and $j^{\text{'}}$ is $\frac{\mathrm{\tau \tau }}{2}$, so that

The dot product of the unit vectors i and $i^{\text{'}}$is localid="1659004543906" $\mathrm{i}.{\mathrm{j}}^{\text{'}}=\left|\mathrm{i}\right|\left|{\mathrm{j}}^{\text{'}}\right|\cos \left(\frac{\mathrm{\tau \tau }}{2}\right)=0$.

And the angle between the axes x and $y^{\text{'}}$of the two unit vectors iand ${\mathrm{j}}^{\text{'}}$is $\frac{\mathrm{\tau \tau }}{2}+\mathrm{\theta }$, so that the dot product of the unit vectors and is localid="1659005132626" $\mathrm{i}.{\mathrm{j}}^{\text{'}}=\left|\mathrm{i}\right|\left|{\mathrm{j}}^{\text{'}}\right|\cos \left(\frac{\mathrm{\tau \tau }}{2}\right)=-\mathrm{sin\theta }$, where $\left|{\mathrm{j}}^{\text{'}}\right|=1,\left|\mathrm{i}\right|=1$, and we use the following relation

$$\begin{gathered} \cos \left(\frac{\mathrm{\tau \tau }}{2}+\mathrm{\theta }\right)=\cos \left(\frac{\mathrm{\tau \tau }}{2}\right)\times \mathrm{cos\theta }-\sin \left(\frac{\mathrm{\tau \tau }}{2}\right)\times \mathrm{sin\theta } \\ =0\times \mathrm{cos\theta }-1\times \mathrm{sin\theta } \\ =-\mathrm{sin\theta } \end{gathered}$$

Also, the angle between the axes y and $y^{\text{'}}$of the two unit vectors j and ${\mathrm{j}}^{\text{'}}$is $\mathrm{\theta }$, so that the dot product of the unit vectors and is localid="1659005460213" $\mathrm{j}.{\mathrm{j}}^{\text{'}}=\left|\mathrm{j}\right|\left|{\mathrm{j}}^{\text{'}}\right|\cos \left(\mathrm{\theta }\right)=\mathrm{cos\theta }$
.

Finally, the dot product of the unit vectors ${\mathrm{i}}^{\text{'}}$and ${\mathrm{i}}^{\text{'}}$must be ${\mathrm{j}}^{\text{'}}.{\mathrm{j}}^{\text{'}}=\left|{\mathrm{j}}^{\text{'}}\right|\left|{\mathrm{j}}^{\text{'}}\right|\cos 0=1$, where

Using the following equation in the equation (2) implies that

$$\begin{gathered} ({\mathrm{i}}^{\text{'}}.{\mathrm{j}}^{\text{'}}){\mathrm{x}}^{\text{'}}+({\mathrm{j}}^{\text{'}}.{\mathrm{j}}^{\text{'}}){\mathrm{y}}^{\text{'}}=(\mathrm{i}.{\mathrm{j}}^{\text{'}})\mathrm{x}+(\mathrm{j}.{\mathrm{j}}^{\text{'}})\mathrm{y} \\ \left(0\right){\mathrm{x}}^{\text{'}}+\left(1\right){\mathrm{y}}^{\text{'}}=-\left(\mathrm{sin\theta }\right)\mathrm{x}+\left(\mathrm{cos\theta }\right)\mathrm{y} \\ {\mathrm{y}}^{\text{'}}=-\left(\mathrm{sin\theta }\right)\mathrm{x}+\left(\mathrm{cos\theta }\right)\mathrm{y} \end{gathered}$$

Therefore, we deduce the following set of equations (1) and (2) are

$$\begin{gathered} {\mathrm{x}}^{\text{'}}=\left(\mathrm{cos\theta }\right)\mathrm{x}+\left(\mathrm{sin\theta }\right)\mathrm{y} \\ {\mathrm{y}}^{\text{'}}=(-\mathrm{sin\theta })\mathrm{x}+\left(\mathrm{cos\theta }\right)\mathrm{y} \end{gathered}$$

Therefore, the axes (x,y) which rotated by an angle $\theta$with respect to the axes $(x^{\text{'}},y^{\text{'}})$obtained from the previous equation which related the components of $\overrightarrow{\mathrm{r}}$and $\overrightarrow{\mathrm{r}}$ written in the matrix form $\left(\begin{array}{c}{x}^{\text{'}}\\ y^{\text{'}}\end{array}\right)=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$,

Axes rotated which represent a passive transformation (vectors not changed but their components changed because the axes are rotated) are verified.