Mathematics: Class 12 (Book 2) — Chapter 5: Three Dimensional Geometry

If a directed line $L$ passing through the origin makes angles $\alpha$, $\beta$, and $\gamma$ with the positive $x$, $y$, and $z$-axes respectively, then:

• The angles $\alpha, \beta, \gamma$ are called the direction angles of the line.
• The cosines of these angles, namely $\cos\alpha$, $\cos\beta$, and $\cos\gamma$, are called the direction cosines (d.c.'s) of the line, typically denoted by $l$, $m$, and $n$:$$l = \cos\alpha, \quad m = \cos\beta, \quad n = \cos\gamma$$
• Fundamental Identity: The sum of the squares of the direction cosines of any line is always equal to unity:$$l^2 + m^2 + n^2 = 1$$
• Reversal of Direction: If the direction of the line is reversed, the direction angles are replaced by their supplements $(\pi - \alpha, \pi - \beta, \pi - \gamma)$, which reverses the signs of the direction cosines to $-l, -m, -n$.

Any three numbers $a$, $b$, and $c$ that are proportional to the direction cosines $l$, $m$, and $n$ of a line are called the direction ratios (d.r.'s) or direction numbers of the line.

• Thus, for some non-zero real number $\lambda$:$$a = \lambda l, \quad b = \lambda m, \quad c = \lambda n$$
• Relationship between d.c.'s and d.r.'s: $$l = \pm \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \pm \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \pm \frac{c}{\sqrt{a^2 + b^2 + c^2}}$$The sign (positive or negative) is chosen consistently depending on the desired direction of the line.

For a unique line passing through two given points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$:

• The distance $PQ$ is given by:$$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
• The direction cosines of the line segment $PQ$ are:$$l = \frac{x_2 - x_1}{PQ}, \quad m = \frac{y_2 - y_1}{PQ}, \quad n = \frac{z_2 - z_1}{PQ}$$
• The direction ratios of the line segment joining $P$ and $Q$ can be taken simply as:$$x_2 - x_1, \quad y_2 - y_1, \quad z_2 - z_1$$

Let the line pass through a point $A$ with position vector $\vec{a}$ and be parallel to a non-zero vector $\vec{b}$.

• Vector Equation: $$\vec{r} = \vec{a} + \lambda \vec{b}$$where $\vec{r}$ is the position vector of any arbitrary point $P(x, y, z)$ on the line, and $\lambda$ is a real parameter.
• Cartesian Equation: If $A$ is $(x_1, y_1, z_1)$ and the direction ratios of the parallel vector $\vec{b}$ are $a, b, c$:$$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$
• If the direction cosines $l, m, n$ of the line are known instead of direction ratios, the equation is:$$\frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n}$$

Let the line pass through two points $A$ and $B$ with position vectors $\vec{a}$ and $\vec{b}$ respectively.

• Vector Equation: $$\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$$
• Cartesian Equation: If the points are $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$:$$\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$$

• Vector Form: If the lines are parallel to vectors $\vec{b}_1$ and $\vec{b}_2$, the acute angle $\theta$ between them is:$$\cos\theta = \left| \frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1| |\vec{b}_2|} \right|$$
• Cartesian Form: If the direction ratios of the two lines are $a_1, b_1, c_1$ and $a_2, b_2, c_2$:$$\cos\theta = \left| \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}} \right|$$
• Using Direction Cosines: If the direction cosines are $l_1, m_1, n_1$ and $l_2, m_2, n_2$:$$\cos\theta = |l_1 l_2 + m_1 m_2 + n_1 n_2|$$

• Perpendicular Lines ($\theta = 90^\circ$): $$a_1 a_2 + b_1 b_2 + c_1 c_2 = 0 \quad \text{or} \quad l_1 l_2 + m_1 m_2 + n_1 n_2 = 0$$
• Parallel Lines ($\theta = 0^\circ$): $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$

The shortest distance between two skew lines is the length of the common perpendicular segment joining them.

• Vector Form: For lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$:$$d = \left| \frac{(\vec{b}_1 \times \vec{b}_2) \cdot (\vec{a}_2 - \vec{a}_1)}{|\vec{b}_1 \times \vec{b}_2|} \right|$$
• Cartesian Form: For lines $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$:$$d = \frac{\begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix}}{\sqrt{(b_1 c_2 - b_2 c_1)^2 + (c_1 a_2 - c_2 a_1)^2 + (a_1 b_2 - a_2 b_1)^2}}$$

If two lines are parallel, they are coplanar and have the same direction vector $\vec{b}$.

• Vector Form: For lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}$:$$d = \left| \frac{\vec{b} \times (\vec{a}_2 - \vec{a}_1)}{|\vec{b}|} \right|$$