Mathematics · Ch 11 — Three-Dimensional Geometry

Summary

Direction Cosines: For a line with angles $\alpha, \beta, \gamma$ to axes, direction cosines are $l = \cos\alpha$ , $m = \cos\beta$ , $n = \cos\gamma$ , with $l^2 + m^2 + n^2 = 1$ .
Direction Ratios: Any triple $(a, b, c)$ proportional to $(l, m, n)$ ; if $a, b, c$ are direction ratios, then $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$ , etc.
Equation of a Line: Through point $(x_1, y_1, z_1)$ with direction ratios $(a, b, c)$ : $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} = \lambda$ (symmetric form).
Angle Between Two Lines: If direction cosines are $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ , then $\cos\theta = |l_1 l_2 + m_1 m_2 + n_1 n_2|$ ; lines are perpendicular if sum $= 0$ , parallel if ratios proportional.
Shortest Distance Between Skew Lines: For lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$ , distance $d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$ .
Equation of a Plane: In normal form: $\vec{r} \cdot \hat{n} = d$ (where $\hat{n}$ is unit normal, $d$ is distance from origin). General form: $ax + by + cz + d = 0$ .
Plane Through Three Points: If points are $(x_1, y_1, z_1)$ , $(x_2, y_2, z_2)$ , $(x_3, y_3, z_3)$ , the equation is $\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0$ .
Angle Between Two Planes: For normals $\vec{n}_1$ and $\vec{n}_2$ , $\cos\theta = \frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$ ; planes are perpendicular if dot product $= 0$ , parallel if normals are proportional. …

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