91Ó°ÊÓ

First find the angles between the vectors$n=2$,$n=3$and$\mathrm{n}=4$.

Consider the solution below.

For$\mathrm{n}=2$.

$$\begin{gathered} {\mathrm{θ}}_2=arc\cos \frac{{\displaystyle \stackrel{âŠ\yen }{u}}.\stackrel{âŠ\yen }{v}}{\left|\left|\stackrel{r}{u}\right|\right|.\left|\left|\stackrel{r}{u}\right|\right|} \\ =arc\cos \frac{{\left(1,1\right)}^T.\left(1,0\right)}{\sqrt{1^2+1^2}\sqrt{1^2+0}} \\ =arc\cos \frac{1}{\sqrt{2}} \\ =45° \end{gathered}$$

Consider the graph for n=2.

For n=3.

$$\begin{gathered} {\mathrm{θ}}_3=arc\cos \frac{{\displaystyle \stackrel{âŠ\yen }{u}.\stackrel{âŠ\yen }{v}}}{\left|\left|\stackrel{r}{u}\right|\right|.\left|\left|\stackrel{r}{u}\right|\right|} \\ =arc\cos \frac{{\left(1,1,1\right)}^T.\left(1,0,0\right)}{\sqrt{1^2+1^2+1^2}\sqrt{1^2+0^2+0^2}} \\ =arc\cos \frac{1}{\sqrt{3}} \\ =54.74° \end{gathered}$$

Consider the graph for n=3.

For n=4.

$$\begin{gathered} {\mathrm{θ}}_4=arc\cos \frac{{\displaystyle \stackrel{âŠ\yen }{u}.\stackrel{âŠ\yen }{v}}}{\left|\left|\stackrel{r}{u}\right|\right|.\left|\left|\stackrel{r}{u}\right|\right|} \\ =arc\cos \frac{{\left(1,1,1,1\right)}^T.\left(1,0,0,0\right)}{\sqrt{1^2+1^2+1^2+1^2}\sqrt{1^2+0^2+0^2+0^2}} \\ =arc\cos \frac{1}{2} \\ =60° \end{gathered}$$

We can see that ${\mathrm{θ}}_n=arc\cos \frac{1}{\sqrt{n}}$. Since, the only thing that changes are the length of the vector $\stackrel{âŠ\yen }{u}$. Let’s find out the limit of $\mathrm{θ}$ when n approaches infinity. Keep in mind that arcos is a continuous function.

Observe the following.

$$\begin{gathered} \mathrm{θ}=\underset{n→∞}{lim}{\mathrm{θ}}_n \\ =\underset{n→∞}{lim}arc\cos \frac{1}{\sqrt{n}} \\ =arc\cos \underset{n→∞}{lim}\frac{1}{\sqrt{n}} \\ =arc\cos \frac{1}{∞} \\ =arc\cos 0 \\ =90° \end{gathered}$$

Hence, the angles between the vectors $\stackrel{âŠ\yen }{u}$and $\stackrel{âŠ\yen }{v}$for $n=2,3,4$ are ${\mathrm{θ}}_2=45°,{\mathrm{θ}}_3=54.74°,{\mathrm{θ}}_4=60°.$

When n approaches infinity, $\left|\left|\stackrel{âŠ\yen }{u}\right|\right|$approaches infinity, and $\mathrm{θ}$approaches$90°$.