Question

Consider three unit vectors ${\stackrel{\perp }{\mathbf{v}}}_1,{\stackrel{\perp }{\mathbf{v}}}_2$, and ${\stackrel{\perp }{\mathbf{v}}}_3$in ${\mathit{R}}^{\mathbf{n}}$. We are told that . What are the possible values of role="math" localid="1660106727111" ${\stackrel{\perp }{\mathbf{v}}}_1,{\stackrel{\perp }{\mathbf{v}}}_2={\stackrel{\perp }{\mathbf{v}}}_1,{\stackrel{\perp }{\mathbf{v}}}_3=\mathbf{1}\mathbf{/}\mathbf{2}$? What could the angle between the vectors ${\stackrel{\perp }{\mathit{v}}}_2$and ${\stackrel{\perp }{\mathit{v}}}_3$ be? Give examples; draw sketches for the cases n = 2 and n = 3 .

Short answer

Angle between the vectors ${\stackrel{\perp }{\mathrm{v}}}_1,{\stackrel{\perp }{\mathrm{v}}}_2$ are $\theta =\left[0,\tau \tau ,2\tau \tau ,....,n\tau \tau \right]$.

For n = 2, graph is,

For n = 3 , the graph is,

Step-by-step solution

Angle between v⊥2 and  v⊥3

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

Thus, we get the following

$$\begin{gathered} {\stackrel{\perp }{v}}_1.{\stackrel{\perp }{v}}_2={\stackrel{\perp }{v}}_1.{\stackrel{\perp }{v}}_3=1/2 \\ \Rightarrow {\stackrel{r}{v}}_1.{\stackrel{r}{v}}_2={\stackrel{r}{v}}_1.{\stackrel{r}{v}}_3 \\ \Rightarrow {\stackrel{r}{v}}_2.{\stackrel{r}{v}}_3 \end{gathered}$$

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

$$\begin{gathered} {\stackrel{\perp }{v}}_2.{\stackrel{\perp }{v}}_3={\stackrel{\perp }{v}}_2.\left({\stackrel{\perp }{v}}_2\right) \\ \Rightarrow {\stackrel{r}{v}}_2.{\stackrel{r}{v}}_3=1 \end{gathered}$$

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

$$\begin{gathered} {\stackrel{\perp }{v}}_2.{\stackrel{\perp }{v}}_3=\left|\left|{\stackrel{\perp }{v}}_2\right|\right|.\parallel {\stackrel{\perp }{v}}_3|cos\theta \\ \Rightarrow {\stackrel{r}{v}}_1.{\stackrel{r}{v}}_2=\left|\left|{\stackrel{r}{v}}_2\right|\right|.║{\stackrel{r}{v}}_3|\cos \theta =1 \end{gathered}$$

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

$$\begin{gathered} {\stackrel{\perp }{v}}_2.{\stackrel{\perp }{v}}_3=\left|\left|{\stackrel{\perp }{v}}_2\right|\right|.\parallel {\stackrel{\perp }{v}}_3|cos\theta =1 \\ \Rightarrow 1\left(1\right)\cos \theta =1 \\ \Rightarrow \cos \theta =1 \end{gathered}$$

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

Consider the graph for  n = 2

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

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

Consider the graph for n =3

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, ${\stackrel{\perp }{v}}_2.{\stackrel{\perp }{v}}_3=1$.

Angle between the ${\stackrel{\perp }{v}}_2$and ${\stackrel{\perp }{v}}_3$is$\theta =\left[0,\tau \tau ,2\tau \tau ,....,n\tau \tau \right]$

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 ,

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