Direction Cosines (d.c.’s) and Direction Ratios (d.r.’s)
1. Direction Cosines:
If a line makes angles α,β,γ with the positive direction of the coordinate axes (x,y, and z), then cosα,cosβ, and cosγ are called the direction cosines of the line.
They are generally denoted by l,m, and n (so l=cosα,m=cosβ,n=cosγ).
Property: The sum of the squares of the direction cosines of any line is one: $$l^2 + m^2 + n^2 = 1.
- **Axes d.c.'s:** The direction cosines of the $x$-axis are $1, 0, 0$; the $y$-axis are $0, 1, 0$; and the $z$-axis are $0, 0, 1$.
**2. Direction Ratios:**
Any quantities that are proportional to the direction cosines of a line are called the direction ratios (d.r.'s) of the line, typically denoted by $a, b,$ and $c$.
- **Relation to d.c.'s:** $$l = \frac{a}{\sqrt{a^2+b^2+c^2}}$$ $$m = \frac{b}{\sqrt{a^2+b^2+c^2}}$$ $$n = \frac{c}{\sqrt{a^2+b^2+c^2}}$$.
- **Two Points:** The direction ratios of a line joining two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are $x_2 - x_1$, $y_2 - y_1$, and $z_2 - z_1$.
**3. Angle Between Two Lines ($\theta$):**
If two lines have direction cosines $l_1, m_1, n_1$ and $l_2, m_2, n_2$, the angle between them is defined by:
$$\cos\theta = l_1 l_2 + m_1 m_2 + n_1 n_2$$.
If they have direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$, the angle is:
$$\cos\theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}$$.
- **Perpendicular Lines ($\theta = 90^\circ$):** $$l_1 l_2 + m_1 m_2 + n_1 n_2 = 0$$, or $$a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$$.
- **Parallel Lines ($\theta = 0^\circ$):** $$\frac{l_1}{l_2} = \frac{m_1}{m_2} = \frac{n_1}{n_2}$$, or $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$.
---
### Plane Geometry Theories and Formulas
**1. Definition and General Equation:**
A plane is a surface such that if any two points are taken on it, the straight line joining them completely lies on the surface.
Every equation of the first degree in $x, y, z$ represents a plane.
- **General Form:** $$ax + by + cz + d = 0$$, where $a, b,$ and $c$ (which are not all zero) represent the direction ratios of the normal to the plane.
- **Special Cases:**
- Passes through the origin if $d = 0$.
- Parallel to $x$-axis if $a = 0$; parallel to $y$-axis if $b = 0$; parallel to $z$-axis if $c = 0$.
**2. Various Forms of a Plane Equation:**
- **Through a given point $(x_1, y_1, z_1)$:** $$a(x - x_1) + b(y - y_1) + c(z - z_1) = 0$$
- **Through three points $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3)$:** Can be found using the determinant equated to zero:
$$\begin{vmatrix} x & y & z & 1 \\ x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \end{vmatrix} = 0$$.
- **Intercept Form:** $$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$$, where $a, b,$ and $c$ are the intercepts cut from the coordinate axes.
- **Normal Form:** $$lx + my + nz = P$$, where $l, m, n$ are the direction cosines of the normal from the origin, and $P$ is the perpendicular distance from the origin.
**3. Distances:**
- **Perpendicular Distance:** The length of the perpendicular distance from a point $(x_1, y_1, z_1)$ to the plane $ax + by + cz + d = 0$ is:
$$\frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$$.
- **Distance to coordinate planes:** From $(x, y, z)$ to the $XY$ plane is $|z|$; to the $YZ$ plane is $|x|$; to the $ZX$ plane is $|y|$.
**4. Angles and Intersections:**
- **Angle Between Two Planes:** It is defined as the angle between their normals. For $a_1x + b_1y + c_1z + d_1 = 0$ and $a_2x + b_2y + c_2z + d_2 = 0$:
$$\cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$$.
- **Perpendicular:** $a_1a_2 + b_1b_2 + c_1c_2 = 0$.
- **Parallel:** $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.
- **Angle Between a Line and a Plane:** For a line with d.r.'s $a_1, b_1, c_1$ and a plane with normal d.r.'s $a_2, b_2, c_2$, the angle $\theta$ is:
$$\sin\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$$.
- If parallel, $a_1a_2 + b_1b_2 + c_1c_2 = 0$.
- If perpendicular, $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.
- **Plane Containing a Line:** A plane $a(x - x_1) + b(y - y_1) + c(z - z_1) = 0$ contains the line $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ if the line's direction is perpendicular to the plane's normal, fulfilling the condition: $a a_1 + b b_1 + c c_1 = 0$.
- **Intersection of Two Planes:** A plane going through the line of intersection of two planes $P_1 = 0$ and $P_2 = 0$ takes the form $$P_1 + k P_2 = 0$$ (i.e., $(a_1x + b_1y + c_1z + d_1) + K(a_2x + b_2y + c_2z + d_2) = 0$).