91影视

Consider the three unit vectors ${\stackrel{鈯�}{v}}_1,{\stackrel{鈯�}{v}}_2,{\stackrel{鈯�}{v}}_3$in $R^n$and if ${\stackrel{鈯�}{v}}_1.{\stackrel{鈯�}{v}}_2={\stackrel{鈯�}{v}}_1.{\stackrel{鈯�}{v}}_3=1/2$.

Thus, we get the following

$$\begin{gathered} {\stackrel{鈯�}{v}}_1.{\stackrel{鈯�}{v}}_2={\stackrel{鈯�}{v}}_1.{\stackrel{鈯�}{v}}_3=1/2 \\ 鈬�{\stackrel{r}{v}}_1.{\stackrel{r}{v}}_2={\stackrel{r}{v}}_1.{\stackrel{r}{v}}_3 \\ 鈬�{\stackrel{r}{v}}_2.{\stackrel{r}{v}}_3 \end{gathered}$$

Now, evaluating ${\stackrel{鈯�}{v}}_2.{\stackrel{鈯�}{v}}_3$as follow

$$\begin{gathered} {\stackrel{鈯�}{v}}_2.{\stackrel{鈯�}{v}}_3={\stackrel{鈯�}{v}}_2.\left({\stackrel{鈯�}{v}}_2\right) \\ 鈬�{\stackrel{r}{v}}_2.{\stackrel{r}{v}}_3=1 \end{gathered}$$

Now, to find the angle between ${\stackrel{鈯�}{v}}_2$and ${\stackrel{鈯�}{v}}_3$consider

$$\begin{gathered} {\stackrel{鈯�}{v}}_2.{\stackrel{鈯�}{v}}_3=\left|\left|{\stackrel{鈯�}{v}}_2\right|\right|.鈭�{\stackrel{鈯�}{v}}_3|cos胃 \\ 鈬�{\stackrel{r}{v}}_1.{\stackrel{r}{v}}_2=\left|\left|{\stackrel{r}{v}}_2\right|\right|.鈺�{\stackrel{r}{v}}_3|\cos 胃=1 \end{gathered}$$

Now, as $\left|\left|{\stackrel{鈯�}{v}}_2\right|\right|.\left|\left|{\stackrel{鈯�}{v}}_3\right|\right|=1$we get the following

$$\begin{gathered} {\stackrel{鈯�}{v}}_2.{\stackrel{鈯�}{v}}_3=\left|\left|{\stackrel{鈯�}{v}}_2\right|\right|.鈭�{\stackrel{鈯�}{v}}_3|cos胃=1 \\ 鈬�1\left(1\right)\cos 胃=1 \\ 鈬�\cos 胃=1 \end{gathered}$$

So, the value of $胃$is as,$胃=\left[0,蟿蟿,2蟿蟿,....,n蟿蟿\right]$ radians

For n = 2 , consider the following examples in $R^2$.

${\stackrel{r}{v}}_2=\left[\begin{array}{c}\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right],{\stackrel{r}{v}}_3=\left[\begin{array}{c}\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right]$

For n = 3 , consider the following examples in $R^3$.

${\stackrel{r}{v}}_2=\left[\begin{array}{c}\frac{\sqrt{2}}{2}\\ 0\\ \frac{\sqrt{2}}{2}\end{array}\right],{\stackrel{r}{v}}_3=\left[\begin{array}{c}\frac{\sqrt{2}}{2}\\ 0\\ \frac{\sqrt{2}}{2}\end{array}\right]$

Hence, ${\stackrel{鈯�}{v}}_2.{\stackrel{鈯�}{v}}_3=1$.

Angle between the ${\stackrel{鈯�}{v}}_2$and ${\stackrel{鈯�}{v}}_3$is$胃=\left[0,蟿蟿,2蟿蟿,....,n蟿蟿\right]$

For n = 2,

${\stackrel{r}{v}}_2=\left[\begin{array}{c}\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right],{\stackrel{r}{v}}_3=\left[\begin{array}{c}\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right]$

For n = 3 ,

${\stackrel{r}{v}}_2=\left[\begin{array}{c}\frac{\sqrt{2}}{2}\\ 0\\ \frac{\sqrt{2}}{2}\end{array}\right],{\stackrel{r}{v}}_3=\left[\begin{array}{c}\frac{\sqrt{2}}{2}\\ 0\\ \frac{\sqrt{2}}{2}\end{array}\right]$