In this question, the space $E$ has dimension $n \geq 2$. Conversely, assume that a pair $(\alpha , \beta)$ of non-collinear vectors of $E$ is found in one of the configurations listed in the table below. Show that the real number $2 \frac { \langle \alpha , \beta \rangle } { \langle \alpha , \alpha \rangle }$ is an integer; specify its value.
$\theta _ { \alpha , \beta }$
$\cos \theta _ { \alpha , \beta }$
$\| \beta \| / \| \alpha \|$
$\pi / 2$
0
$\geq 1$
$\pi / 3$
$1 / 2$
1
$2 \pi / 3$
$- 1 / 2$
1
$\pi / 4$
$\sqrt { 2 } / 2$
$\sqrt { 2 }$
$3 \pi / 4$
$- \sqrt { 2 } / 2$
$\sqrt { 2 }$
$\pi / 6$
$\sqrt { 3 } / 2$
$\sqrt { 3 }$
$5 \pi / 6$
$- \sqrt { 3 } / 2$
$\sqrt { 3 }$
In this question, the space $E$ has dimension $n \geq 2$. Conversely, assume that a pair $(\alpha , \beta)$ of non-collinear vectors of $E$ is found in one of the configurations listed in the table below. Show that the real number $2 \frac { \langle \alpha , \beta \rangle } { \langle \alpha , \alpha \rangle }$ is an integer; specify its value.
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
$\theta _ { \alpha , \beta }$ & $\cos \theta _ { \alpha , \beta }$ & $\| \beta \| / \| \alpha \|$ \\
\hline
$\pi / 2$ & 0 & $\geq 1$ \\
$\pi / 3$ & $1 / 2$ & 1 \\
$2 \pi / 3$ & $- 1 / 2$ & 1 \\
$\pi / 4$ & $\sqrt { 2 } / 2$ & $\sqrt { 2 }$ \\
$3 \pi / 4$ & $- \sqrt { 2 } / 2$ & $\sqrt { 2 }$ \\
$\pi / 6$ & $\sqrt { 3 } / 2$ & $\sqrt { 3 }$ \\
$5 \pi / 6$ & $- \sqrt { 3 } / 2$ & $\sqrt { 3 }$ \\
\hline
\end{tabular}
\end{center}