3D Geometry Concepts and Formulas

Vocabulary flashcards covering core 3D geometry definitions, direction cosines, internal division, and line distances from the lecture notes.

Direction Cosines (DCs) of Vector $\mathbf{A}$

The ratios $l = \frac{a}{|\mathbf{A}|}$, $m = \frac{b}{|\mathbf{A}|}$, and $n = \frac{c}{|\mathbf{A}|}$ for a vector $\mathbf{A} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$, which satisfy the identity $l^2 + m^2 + n^2 = 1$ and can also be expressed as $-l, -m, -n$.

Direction Angles Identity for Sine

The trigonometric identity $\sin^2(\alpha) + \sin^2(\beta) + \sin^2(\gamma) = 2$, where $\alpha$, $\beta$, and $\gamma$ are the direction angles made by a line or vector with the coordinate axes.

Internal Division Formula in 3D

The coordinates of point $P$ dividing the line segment between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ internally in the ratio $m:n$, given by $\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right)$.

Angle Between Two Lines (Direction Cosines)

The cosine of the angle $\theta$ between two lines with direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$, given by $\cos(\theta) = l_1 l_2 + m_1 m_2 + n_1 n_2$.

Angle Between Two Lines (Direction Ratios)

The cosine of the angle $\theta$ between two lines with direction ratios $\mathbf{DR}_1$ and $\mathbf{DR}_2$, given by $\cos(\theta) = \frac{\mathbf{DR}_1 \cdot \mathbf{DR}_2}{|\mathbf{DR}_1| |\mathbf{DR}_2|}$.

Distance Between Parallel Lines

The shortest distance $d$ between two parallel lines $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$ and $\mathbf{r} = \mathbf{c} + \mu \mathbf{b}$, given by $d = \frac{|(\mathbf{c} - \mathbf{a}) \times \mathbf{b}|}{|\mathbf{b}|}$.

Distance Between Skew Lines

The shortest distance $d$ between two non-parallel, non-intersecting lines $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$ and $\mathbf{r} = \mathbf{c} + \mu \mathbf{d}$, given by $d = \frac{|[\mathbf{a} - \mathbf{c}, \mathbf{b}, \mathbf{d}]|}{|\mathbf{b} \times \mathbf{d}|}$.