Three Dimensional Geometry Flashcards

This set of vocabulary flashcards covers key formulas and definitions for Three Dimensional Geometry, including direction cosines, direction ratios, and equations of lines in space.

Direction Cosines Identity

The relationship between the direction cosines $l, m, n$ of a line, expressed as $l^2 + m^2 + n^2 = 1$.

Direction Ratios (D.R's)

Three numbers $a, b, c$ that are proportional to the direction cosines $l, m, n$ of a line.

Formula for D.C's given D.R's

The direction cosines are calculated as l = \frac{\text{±}a}{\text{\sqrt{a^2 + b^2 + c^2}}}, m = \frac{\text{±}b}{\text{\sqrt{a^2 + b^2 + c^2}}}, and n = \frac{\text{±}c}{\text{\sqrt{a^2 + b^2 + c^2}}}.

D.R's of a line passing through two points

For points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$, the direction ratios are $x_2 - x_1, y_2 - y_1, z_2 - z_1$.

Vector form of a line equation

The equation of a line passing through a point with position vector $\mathbf{a}$ and parallel to a vector $\mathbf{b}$, given as $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$.

Cartesian form of a line equation

The representation of a line through point $(x_1, y_1, z_1)$ with direction ratios $a, b, c$ as $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$.

Cosine angle formula using D.C's

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

Condition for perpendicular lines

Two lines are perpendicular if the sum of the products of their direction cosines is zero, i.e., $l_1 l_2 + m_1 m_2 + n_1 n_2 = 0$.

Condition for parallel lines

Two lines are parallel if their direction ratios are proportional, such that $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

Angle between two lines formula (D.R's)

The cosine of the angle $\theta$ between lines with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ is $\cos(\theta) = \left| \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \right|$.