91影视

Given, ${\mathrm{u}}_1$ is a unit vector in the direction of an incident ray of light, and ${\mathrm{u}}_3$ and ${\mathrm{u}}_2$ are unit vectors in the directions of the reflected and refracted rays and if is a unit vector normal to the surface$\mathrm{AB}$ , the laws of optics say that${\mathrm{胃}}_1={\mathrm{胃}}_3$ and ${\mathrm{n}}_1{\mathrm{蝉颈苍胃}}_1={\mathrm{n}}_2{\mathrm{蝉颈苍胃}}_2$, where ${\mathrm{n}}_1$ and ${\mathrm{n}}_2$ are constants.

The triple scalar product for three vectors is written as $\mathbf{A}\mathbf{.}\left(B脳C\right)$ .

The determinant is as follows.

$\mathbf{A}\mathbf{.}\left(B脳C\right)\mathbf{=}\left|\begin{array}{ccc}{A}_x& A_y& A_z\\ B_x& B_y& B_z\\ C_x& C_y& C_z\end{array}\right|$

The vectors$\mathrm{u}$, ${\mathrm{u}}_1$,${\mathrm{u}}_2$,${\mathrm{u}}_3$are unit vectors. The magnitude of these vectors are equal to one.

${\mathrm{u}}_1$is the incident ray and ${\mathrm{u}}_3$ is the reflected ray.

Both are in opposite directions. Even their distance is equal, their directions are opposite. So,${\mathrm{u}}_1=鈥�{\mathrm{u}}_3$

The angles are calculated as below.

$$\begin{gathered} {\mathrm{u}}_1.\mathrm{u}=\left|{\mathrm{u}}_1\right|\left|\mathrm{u}\right|\mathrm{肠辞蝉胃} \\ {\mathrm{u}}_3.\mathrm{u}=\left|{\mathrm{u}}_3\right|\left|\mathrm{u}\right|{\mathrm{肠辞蝉胃}}_3 \end{gathered}$$

Since, ${\mathrm{u}}_1=鈥�{\mathrm{u}}_3$

$\left|{\mathrm{u}}_1\right|\left|\mathrm{u}\right|{\mathrm{肠辞蝉胃}}_1=\left|鈥�{\mathrm{u}}_3\right|\left|\mathrm{u}\right|{\mathrm{肠辞蝉胃}}_3$

So, ${\mathrm{胃}}_1={\mathrm{胃}}_3$can be written as ${\mathrm{u}}_1.\mathrm{u}=鈥�{\mathrm{u}}_3.\mathrm{u}$.

The incident and the reflected rays are mentioned as follows.

${\mathrm{u}}_1$is the incident ray and ${\mathrm{u}}_2$is the reflected ray.

$$\begin{gathered} {\mathrm{u}}_1脳\mathrm{u}\left|{\mathrm{u}}_1\right|\left|\mathrm{u}\right|{\mathrm{蝉颈苍胃}}_1={\mathrm{蝉颈苍胃}}_1 \\ {\mathrm{u}}_2脳\mathrm{u}\left|{\mathrm{u}}_2\right|\left|\mathrm{u}\right|{\mathrm{蝉颈苍胃}}_2=\mathrm{蝉颈苍胃}2 \end{gathered}$$

Since, ${\mathrm{n}}_1{\mathrm{蝉颈苍胃}}_1={\mathrm{n}}_2{\mathrm{蝉颈苍胃}}_2$

${\mathrm{n}}_1{\mathrm{u}}_1脳\mathrm{u}={\mathrm{n}}_2{\mathrm{u}}_2脳\mathrm{u}$

Hence, the laws in dot product and vector product are ${\mathrm{u}}_{1.}\mathrm{u}=鈥�{\mathrm{u}}_3.\mathrm{u}$ and ${\mathrm{n}}_1{\mathrm{u}}_1脳\mathrm{u}={\mathrm{n}}_2{\mathrm{u}}_2脳\mathrm{u}$ respectively.