Geometry and Trigonometry

In geometry, lines can have different spatial relationships with each other. When two lines occupy the same space, they are termed coincident. This means that every point on one line is also a point on the other line, and they are essentially the same line. Parallel lines, on the other hand, are two lines that are always equidistant from each other and will never intersect. In two-dimensional geometry, parallel lines can be recognized by having the same slope. Mathematically, if we have two lines \(L_1: y = m_1x + b_1\) and \(L_2: y = m_2x + b_2\), these lines are parallel if and only if \(m_1 = m_2\) and \(b_1 \neq b_2\).

Intersecting lines are lines that cross each other at a single point, known as the point of intersection. For two lines defined by \(L_1: y = m_1x + b_1\) and \(L_2: y = m_2x + b_2\), if \(m_1 \neq m_2\), then the lines will intersect. The point of intersection can be found by solving the system of equations for \(x\) and \(y\). Lastly, in three-dimensional space, we have skew lines. Skew lines are neither parallel nor intersecting, and do not lie in the same plane. Imagine two lines that are at different heights and do not meet; these are skew lines. The equations of lines in three-dimensional space can be represented parametrically as \(L_1: \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix} + t_1 \begin{pmatrix} a_1 \\ b_1 \\ c_1 \end{pmatrix}\) and \(L_2: \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix} + t_2 \begin{pmatrix} a_2 \\ b_2 \\ c_2 \end{pmatrix}\). They are skew if they do not intersect and are not parallel.