Three-Dimensional Geometry - Direction Cosines and Ratios of a Line

Direction Angles: A line in three-dimensional space makes specific angles $\alpha$, $\beta$, and $\gamma$ with the positive directions of the $X$, $Y$, and $Z$ axes respectively. These are called direction angles. Visually, these angles quantify the 'tilt' of the line relative to the three fixed reference axes in a Cartesian coordinate system, with the vertex of the angles effectively at the point where the line (or a parallel line) passes through the origin.

Direction Cosines (DCs): The cosines of the direction angles, denoted as $l = \cos\alpha$, $m = \cos\beta$, and $n = \cos\gamma$, are known as direction cosines. A fundamental property is that $l^2 + m^2 + n^2 = 1$. Visually, these can be thought of as the $(x, y, z)$ coordinates of a point on a unit sphere centered at the origin, representing the direction vector of the line.

Direction Ratios (DRs): Any three numbers $a$, $b$, and $c$ that are proportional to the direction cosines $l$, $m$, and $n$ are called direction ratios. Thus, $\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = k$. Unlike direction cosines, which are unique for a directed line, direction ratios can be any set of numbers representing the same direction. Visually, any vector $\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}$ that lies along or is parallel to the line provides a set of direction ratios $(a, b, c)$.

Line through Two Points: If a line passes through two distinct points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, its direction ratios are found by the differences of their coordinates: $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$. Visually, this is the displacement vector from $P_1$ to $P_2$, indicating how many units one must travel along each axis to move from one point to the other.

Relationship between DCs and DRs: To convert direction ratios $(a, b, c)$ into direction cosines $(l, m, n)$, we divide each ratio by the magnitude of the vector, $\sqrt{a^2 + b^2 + c^2}$. This results in $l = \pm \frac{a}{\sqrt{a^2+b^2+c^2}}$, $m = \pm \frac{b}{\sqrt{a^2+b^2+c^2}}$, and $n = \pm \frac{c}{\sqrt{a^2+b^2+c^2}}$. The $\pm$ sign indicates that a line has two sets of direction cosines depending on the chosen direction of travel along the line.

Angle Between Two Lines: The angle $\theta$ between two lines with direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ is given by $\cos\theta = |l_1l_2 + m_1m_2 + n_1n_2|$. Visually, this is the dot product of the two unit vectors representing the directions of the lines. If the lines are given by direction ratios, the formula incorporates the magnitudes of the corresponding vectors.

Conditions for Parallelism and Perpendicularity: Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are parallel if their ratios are proportional, i.e., $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$. They are perpendicular if the sum of the products of their corresponding ratios is zero, i.e., $a_1a_2 + b_1b_2 + c_1c_2 = 0$. Visually, perpendicular lines are at $90^\circ$ to each other, meaning their direction vectors are orthogonal.