Three Dimensional Geometry Flashcards

Direction Cosines and Direction Ratios of a Line

In three-dimensional geometry, the orientation of a line is defined using Direction Cosines (DCs) and Direction Ratios (DRs). Direction Cosines are represented by the symbols $l$, $m$, and $n$. These represent the cosines of the angles $\alpha$, $\beta$, and $\gamma$ that a line makes with the positive directions of the $x$, $y$, and $z$ axes, respectively. Mathematically, this is expressed as $l = \cos(\alpha)$, $m = \cos(\beta)$, and $n = \cos(\gamma)$. A fundamental property connecting these three values is that the sum of their squares is always equal to unity, stated by the formula $l^2 + m^2 + n^2 = 1$.

Direction Ratios are defined as any three numbers $a$, $b$, and $c$ that are proportional to the Direction Cosines of a line. This proportionality implies that $\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = \lambda$, where $\lambda$ is a constant. Using the identity $l^2 + m^2 + n^2 = 1$, we can derive that $(a\lambda)^2 + (b\lambda)^2 + (c\lambda)^2 = 1$. Solving for $\lambda$ yields $\lambda^2(a^2 + b^2 + c^2) = 1$, which means $\lambda = \pm \frac{1}{\sqrt{a^2 + b^2 + c^2}}$. Consequently, the Direction Cosines can be calculated from the Direction Ratios using 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 $n = \pm \frac{c}{\sqrt{a^2 + b^2 + c^2}}$.

Direction Cosines of a Line Passing Through Two Points

Consider a line passing through two distinct points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$. To find the Direction Cosines of the line $PQ$, we first determine the Direction Ratios, which are simply the differences between the coordinates of the two points: $x_2 - x_1$, $y_2 - y_1$, and $z_2 - z_1$. Alternatively, one can use $x_1 - x_2$, $y_1 - y_2$, and $z_1 - z_2$.

The length of the line segment $PQ$, which serves as the normalizing factor (hypotenuse), is given by the distance formula: $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$. The Direction Cosines are then defined as the ratio of the coordinate differences to this length:

$l = \cos(\alpha) = \frac{x_2 - x_1}{PQ}$

$m = \cos(\beta) = \frac{y_2 - y_1}{PQ}$

$n = \cos(\gamma) = \frac{z_2 - z_1}{PQ}$

Examples of Direction Cosines and Ratios

Example 1 (Finding DCs from Angles): If a line makes angles $90^{\circ}$, $60^{\circ}$, and $30^{\circ}$ with the positive directions of the $x$, $y$, and $z$ axes respectively, find its Direction Cosines. Given $\alpha = 90^{\circ}$, $\beta = 60^{\circ}$, and $\gamma = 30^{\circ}$, we calculate:

• $l = \cos(90^{\circ}) = 0$
• $m = \cos(60^{\circ}) = \frac{1}{2}$
• $n = \cos(30^{\circ}) = \frac{\sqrt{3}}{2}$ The Direction Cosines of the line are $\langle 0, \frac{1}{2}, \frac{\sqrt{3}}{2} \rangle$.

Example 2 (Finding DCs from DRs): If a line has Direction Ratios $2, -1, -2$, determine its Direction Cosines. First, calculate $\sqrt{a^2 + b^2 + c^2} = \sqrt{2^2 + (-1)^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$. The Direction Cosines are:

• $l = \pm \frac{2}{3}$
• $m = \pm \frac{-1}{3}$
• $n = \pm \frac{-2}{3}$ The Direction Cosines are $\pm (\frac{2}{3}, -\frac{1}{3}, -\frac{2}{3})$.

Geometry of a Triangle (BC and AC Calculation): From a specific geometric exercise, for a line segment $BC$ with Direction Ratios $(-4, -6, 4)$, the magnitude is $|BC| = \sqrt{16 + 36 + 16} = \sqrt{68} = 2\sqrt{17}$. The resulting Direction Cosines are $\langle -\frac{2}{\sqrt{17}}, -\frac{3}{\sqrt{17}}, \frac{2}{\sqrt{17}} \rangle$. For a line segment $AC$ passing through points $(3, 5, -4)$ and $(-5, -5, -2)$, the Direction Ratios are $(-8, -10, 2)$. The magnitude is $|AC| = \sqrt{64 + 100 + 4} = \sqrt{168} = 2\sqrt{42}$. The Direction Cosines are $\langle -\frac{8}{2\sqrt{42}}, -\frac{10}{2\sqrt{42}}, \frac{2}{2\sqrt{42}} \rangle$.

Equation of a Line in Space

A line in three-dimensional space can be uniquely determined if we know a point through which it passes and the direction along which it lies (represented by a parallel vector).

Vector Form: Let a line pass through a point $A$ with position vector $\mathbf{a}$ and be parallel to a vector $\mathbf{b}$. If $\mathbf{r}$ is the position vector of any arbitrary point $P(x, y, z)$ on the line, the vector equation is given by $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$, where $\lambda$ is a scalar.

Cartesian Form: If the point $A$ has coordinates $(x_1, y_1, z_1)$ and the parallel vector $\mathbf{b}$ has Direction Ratios $a$, $b$, and $c$ (so $\mathbf{b} = a\hat{i} + b\hat{j} + c\hat{k}$), the equations of the line can be written as $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$.

Example 6: Find the vector and the Cartesian equations of the line through the point $(3, 2, -4)$ and which is parallel to the vector $3\hat{i} + 2\hat{j} - 8\hat{k}$. Note: In the provided worked solution, the starting point was taken as $A(5, 2, -4)$.

• Position vector $\mathbf{a} = 5\hat{i} + 2\hat{j} - 4\hat{k}$.
• Parallel vector $\mathbf{b} = 3\hat{i} + 2\hat{j} - 8\hat{k}$.
• Vector Equation: $\mathbf{r} = (5\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 8\hat{k})$.
• Cartesian Equation: $\frac{x - 5}{3} = \frac{y - 2}{2} = \frac{z + 4}{-8}$.

Angle Between Two Lines

The angle $\theta$ between two lines can be determined using their Direction Ratios or Direction Cosines. Let line $L_1$ have Direction Ratios $a_1, b_1, c_1$ and line $L_2$ have Direction Ratios $a_2, b_2, c_2$.

Using Direction Ratios: The cosine of the angle is given by the formula: $\cos(\theta) = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$ The sine of the angle can be expressed as: $\sin(\theta) = \frac{\sqrt{(a_1b_2 - a_2b_1)^2 + (b_1c_2 - b_2c_1)^2 + (c_1a_2 - c_2a_1)^2}}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$

Using Direction Cosines: If the Direction Cosines of the lines are $\langle l_1, m_1, n_1 \rangle$ and $\langle l_2, m_2, n_2 \rangle$, the formula simplifies significantly because $l^2 + m^2 + n^2 = 1$: $\cos(\theta) = |l_1l_2 + m_1m_2 + n_1n_2|$

Conditions for Perpendicularity and Parallelism:

1. Perpendicularity: Two lines are perpendicular if the angle $\theta = 90^{\circ}$. In this case, $\cos(90^{\circ}) = 0$, leading to the condition: $a_1a_2 + b_1b_2 + c_1c_2 = 0$ or $l_1l_2 + m_1m_2 + n_1n_2 = 0$.
2. Parallelism: Two lines are parallel if the angle $\theta = 0^{\circ}$. This occurs when their Direction Ratios are proportional: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k$.