| Version 5 |
Version 4 |
| The {\em angle between two lines} in a plane is defined to be |
The {\em angle between two lines} in a plane is defined to be |
| \begin{itemize} |
\begin{itemize} |
| \item 0, if the lines are parallel; |
\item 0, if the lines are parallel; |
| \item the smaller angle having as sides the half-lines starting from the intersection point of the lines and lying on those two lines, if the lines are not parallel. |
\item the smaller angle having as sides the half-lines starting from the intersection point of the lines and lying on those two lines, if the lines are not parallel. |
| \end{itemize} |
\end{itemize} |
| Hence, the angle $\theta$ satisfies always |
Hence, the angle $\theta$ satisfies always |
| \begin{align} |
\begin{align} |
| 0 \leqq \theta \leqq \frac{\pi}{2}. |
0 \leqq \theta \leqq \frac{\pi}{2}. |
| \end{align} |
\end{align} |
| If the slopes of the two lines are $m_1$ and $m_2$, the angle $\theta$ is obtained from |
If the slopes of the two lines are $m_1$ and $m_2$, the angle $\theta$ is obtained from |
| $$\tan\theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|.$$ |
$$\tan\theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|.$$ |
| This equation clicks in the case that the lines are perpendicular; then $\theta$ is $\displaystyle\frac{\pi}{2}$. |
This equation clicks in the case that the lines are perpendicular; then $\theta$ is $\displaystyle\frac{\pi}{2}$. |
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|
| In the Euclidean space, the angle $\theta$ between two lines is most comfortably defined by using the direction vectors $\vec{u}$ and $\vec{v}$ of the lines: |
In the Euclidean space, the angle $\theta$ between two lines is most comfortably defined by using the direction vectors $\vec{u}$ and $\vec{v}$ of the lines: |
| $$\cos\theta = \left|\frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}\right|.$$ |
$$\cos\theta = \left|\frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}\right|.$$ |
| Also this angle satisfies (1). |
Also this angle satisfies (1). |
