Key Points

If a line makes angles $\alpha, \beta, \gamma$ with the positive x, y, and z-axes respectively, its direction cosines (d.c.'s) are $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$. They always satisfy the relation $l^2 + m^2 + n^2 = 1$.

Any three numbers $a, b, c$ which are proportional to the direction cosines $l, m, n$ of a line are called its direction ratios (d.r.'s). This means $\frac{l}{a} = \frac{m}{b} = \frac{n}{c}$ for some constant.

If $a, b, c$ are the direction ratios of a line, then its direction cosines are given by the formulas: $l = \pm \frac{a}{\sqrt{a^2+b^2+c^2}}$, $m = \pm \frac{b}{\sqrt{a^2+b^2+c^2}}$, and $n = \pm \frac{c}{\sqrt{a^2+b^2+c^2}}$.

The direction ratios of the line segment joining points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ can be taken as $(x_2 - x_1), (y_2 - y_1), (z_2 - z_1)$.

The vector equation of a line passing through a point with position vector $\vec{a}$ and parallel to a given vector $\vec{b}$ is $\vec{r} = \vec{a} + \lambda \vec{b}$, where $\lambda$ is a scalar parameter.

The Cartesian equation of a line passing through a point $(x_1, y_1, z_1)$ and having direction ratios $a, b, c$ is $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$.

The angle $\theta$ between two lines with direction vectors $\vec{b}_1, \vec{b}_2$ or direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ is given by $\cos \theta = \left|\frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1||\vec{b}_2|}\right| = \left|\frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right|$.

Two lines are perpendicular if their direction vectors have a dot product of zero, i.e., $\vec{b}_1 \cdot \vec{b}_2 = 0$ or $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$. They are parallel if their direction ratios are proportional, i.e., $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

The shortest distance $d$ between two skew lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$ is given by the formula $d = \left|\frac{(\vec{b}_1 \times \vec{b}_2) \cdot (\vec{a}_2 - \vec{a}_1)}{|\vec{b}_1 \times \vec{b}_2|}\right|$.

Two lines intersect if the shortest distance between them is zero. This occurs when the scalar triple product is zero: $(\vec{b}_1 \times \vec{b}_2) \cdot (\vec{a}_2 - \vec{a}_1) = 0$.

The distance $d$ between two parallel lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}$ is given by $d = \left|\frac{\vec{b} \times (\vec{a}_2 - \vec{a}_1)}{|\vec{b}|}\right|$.

The vector equation of a plane at a perpendicular distance $d$ from the origin with unit normal vector $\hat{n}$ is $\vec{r} \cdot \hat{n} = d$. Its Cartesian form is $lx + my + nz = d$, where $l, m, n$ are the direction cosines of the normal.

The equation of a plane passing through a point with position vector $\vec{a}$ and perpendicular to a vector $\vec{N}$ is $(\vec{r} - \vec{a}) \cdot \vec{N} = 0$. The general Cartesian form is $Ax + By + Cz + D = 0$, where $A, B, C$ are direction ratios of the normal.

The vector equation of a plane passing through three non-collinear points with position vectors $\vec{a}, \vec{b}, \vec{c}$ is given by $(\vec{r} - \vec{a}) \cdot [(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})] = 0$.

The angle $\theta$ between two planes is the angle between their normal vectors $\vec{n}_1$ and $\vec{n}_2$. It is calculated using $\cos \theta = \left|\frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1||\vec{n}_2|}\right|$.

The angle $\theta$ between a line with direction vector $\vec{b}$ and a plane with normal vector $\vec{n}$ is given by $\sin \theta = \left|\frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|}\right|$. Note that this uses sine, not cosine.

The perpendicular distance of a point $(x_1, y_1, z_1)$ from the plane $Ax + By + Cz + D = 0$ is given by the formula $d = \left|\frac{Ax_1 + By_1 + Cz_1 + D}{\sqrt{A^2 + B^2 + C^2}}\right|$.

The equation of a plane that makes intercepts $a, b, c$ on the x, y, and z-axes respectively is given by the formula $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$.