Three-Dimensional Geometry: Lines notes: ISC Class 12 Mathematics

1. Direction cosines and direction ratios

If a line makes angles $\alpha$ , $\beta$ , $\gamma$ with the axes, its direction cosines are $l = \cos\alpha$ , $m = \cos\beta$ , $n = \cos\gamma$ , and $l^2 + m^2 + n^2 = 1$ .
Direction ratios $a, b, c$ are any numbers proportional to $l, m, n$ : $l = \dfrac{a}{\sqrt{a^2 + b^2 + c^2}}$ , and so on.
Through $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ : ratios $x_2 - x_1$ , $y_2 - y_1$ , $z_2 - z_1$ ; divide by $PQ$ for the cosines.

$OP$ makes angles $\alpha$, $\beta$, $\gamma$ with the axes: $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$ (sketch, not to scale).

Many sets of ratios describe one line; dividing by the length gives the cosines.
Direction ratios	Length	Direction cosines
$6, -3, 2$	$7$	$\dfrac{6}{7}, -\dfrac{3}{7}, \dfrac{2}{7}$
$12, -6, 4$	$14$	$\dfrac{6}{7}, -\dfrac{3}{7}, \dfrac{2}{7}$
$-6, 3, -2$	$7$	$-\dfrac{6}{7}, \dfrac{3}{7}, -\dfrac{2}{7}$

Subtract, measure, divide, check.
Step	Working
Ratios	$5 - 3,\ 4 - (-2),\ -2 - 1 = 2, 6, -3$
Length	$\sqrt{4 + 36 + 9} = 7$
Cosines	$\dfrac{2}{7}, \dfrac{6}{7}, -\dfrac{3}{7}$
Check	$\dfrac{4 + 36 + 9}{49} = 1$

Formulas

Direction cosines
$l^2 + m^2 + n^2 = 1$

Key terms

Direction cosines: Cosines of the angles a line makes with the three axes.
Direction ratios: Any three numbers proportional to the direction cosines.

2. Coordinate axes and coordinate planes

$x$ -axis: $y = 0$ , $z = 0$ ; $y$ -axis: $x = 0$ , $z = 0$ ; $z$ -axis: $x = 0$ , $y = 0$ .
Vector forms: $\vec{r} = \lambda\hat{i}$ , $\vec{r} = \lambda\hat{j}$ , $\vec{r} = \lambda\hat{k}$ .
Coordinate planes: $xy$ -plane $z = 0$ ; $yz$ -plane $x = 0$ ; $zx$ -plane $y = 0$ .
An axis needs two equations; a coordinate plane needs one.

An axis needs two equations; a coordinate plane needs one.
Object	Equation(s)	Direction cosines
$x$ -axis	$y = 0$ , $z = 0$	$1, 0, 0$
$y$ -axis	$x = 0$ , $z = 0$	$0, 1, 0$
$z$ -axis	$x = 0$ , $y = 0$	$0, 0, 1$
$xy$ -plane	$z = 0$	—
$yz$ -plane	$x = 0$	—
$zx$ -plane	$y = 0$	—

On the $z$-axis, $x = 0$ and $y = 0$ (sketch, not to scale).

Formulas

Example
$\text{z-axis: } x = 0,\ y = 0$

Key terms

Coordinate plane: A plane containing two of the axes, such as $z = 0$ .

3. Angle between two lines

For direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ :

$\cos\theta = \dfrac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}$ (acute angle).

Perpendicular: $a_1a_2 + b_1b_2 + c_1c_2 = 0$ .
Parallel: $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$ .
A direction perpendicular to two lines: $\vec{b}_1 \times \vec{b}_2$ .

Only the directions matter for angles.
Condition on ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$	Meaning
$\cos\theta = \dfrac{\|a_1a_2 + b_1b_2 + c_1c_2\|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}$	acute angle $\theta$ between the lines
$a_1a_2 + b_1b_2 + c_1c_2 = 0$	perpendicular
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$	parallel

$\theta = \cos^{-1}\dfrac23$.
Quantity	Value
Ratios	$1, 2, 2$ and $4, 0, 3$
Sum of products	$4 + 0 + 6 = 10$
Lengths	$3 \text{ and } 5$
$\cos\theta$	$\dfrac{10}{15} = \dfrac23$

Zero sum of products, or equal ratios.
Test	Working	Result
Perpendicular	$2(3) + (-1)(2) + k(-1) = 0$	$k = 4$
Parallel	$\dfrac{-4}{2} = \dfrac{2}{-1} = \dfrac{-6}{3} = -2$	parallel

Formulas

Angle between lines
$\cos\theta = \dfrac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1|\,|\vec{b}_2|}$

Key terms

Perpendicular lines: Lines whose direction ratios have zero sum of products.

4. Equations of a line

Through $\vec{a}$ , parallel to $\vec{b}$ : $\vec{r} = \vec{a} + \lambda\vec{b}$ ; Cartesian $\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}$ .
Through two points $\vec{a}$ , $\vec{b}$ : $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$ ; Cartesian $\dfrac{x - x_1}{x_2 - x_1} = \dfrac{y - y_1}{y_2 - y_1} = \dfrac{z - z_1}{z_2 - z_1}$ .
Standard form: make the coefficients of $x$ , $y$ , $z$ equal to $1$ before reading the point and the ratios.
A general point of the line is $(x_1 + a\lambda,\ y_1 + b\lambda,\ z_1 + c\lambda)$ .

$\vec{r} = \vec{a} + \lambda\vec{b}$ (sketch, not to scale).

Vector form to Cartesian form in two steps.

Direction from the difference of the two points.
Form	Equation
Vector	$\vec{r} = 2\hat{i} - \hat{j} + 4\hat{k} + \lambda(3\hat{i} + 2\hat{j} - 4\hat{k})$
Cartesian	$\dfrac{x - 2}{3} = \dfrac{y + 1}{2} = \dfrac{z - 4}{-4}$
Check with $\lambda = 2$	$(8, 3, -4)$

Make each coefficient of $x$, $y$, $z$ equal to $1$ before reading.
Given part	Standard form	Read off
$\dfrac{3 - x}{4}$	$\dfrac{x - 3}{-4}$	point $3$ , ratio $-4$
$\dfrac{2y + 1}{6}$	$\dfrac{y + \frac12}{3}$	point $-\dfrac12$ , ratio $3$
$\dfrac{z + 2}{-3}$	$\dfrac{z + 2}{-3}$	point $-2$ , ratio $-3$

Formulas

Vector form
$\vec{r} = \vec{a} + \lambda\vec{b}$

Key terms

Direction vector: The vector $\vec{b}$ in $\vec{r} = \vec{a} + \lambda\vec{b}$ .

5. Coplanar and skew lines; intersection; distance from a line

Lines $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$ with non-parallel directions are coplanar (and intersect) iff $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0$ ; otherwise they are skew.
Point of intersection: equate general points, solve two equations for $\lambda$ and $\mu$ , check the third.
Foot of the perpendicular from $P$ : take $F$ on the line with $\vec{PF} \cdot \vec{b} = 0$ ; distance $= PF$ , or $\dfrac{|(\vec{P} - \vec{a}) \times \vec{b}|}{|\vec{b}|}$ .
Image of $P$ in the line: $2F - P$ .

Intersect, parallel or skew?

Two equations fix the parameters; the third decides.
Step	Working
General points	$(1 + 2\lambda,\ \lambda,\ 2 - \lambda)$ and $(4 + \mu,\ 4 + 3\mu,\ 2 + \mu)$
From $x$ and $y$	$2\lambda - \mu = 3,\ \lambda - 3\mu = 4$
Solve	$\lambda = 1,\ \mu = -1$
Check $z$	$2 - 1 = 1 \text{ and } 2 + (-1) = 1$
Point	$(3, 1, 1)$

Foot of the perpendicular $F$ and distance $PF$ (sketch, not to scale).

Make $\vec{PF}$ perpendicular to the direction.
Step	Working
General point	$F = (\lambda,\ 2 - 2\lambda,\ -1 + 2\lambda)$
$\vec{PF}$	$(\lambda - 7,\ 5 - 2\lambda,\ 4 + 2\lambda)$
Perpendicular	$(\lambda - 7) - 2(5 - 2\lambda) + 2(4 + 2\lambda) = 9\lambda - 9 = 0$
Foot	$\lambda = 1,\ F = (1, 0, 1)$
Distance	$PF = \sqrt{36 + 9 + 36} = 9$

Formulas

Coplanar
$(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0$

Key terms

Skew lines: Lines that are not parallel and do not meet.
Foot of the perpendicular: The point of the line nearest to the given point.

6. Shortest distance

Skew lines: $d = \dfrac{|(\vec{b}_1 \times \vec{b}_2) \cdot (\vec{a}_2 - \vec{a}_1)|}{|\vec{b}_1 \times \vec{b}_2|}$ .
Parallel lines $\vec{r} = \vec{a}_1 + \lambda\vec{b}$ , $\vec{r} = \vec{a}_2 + \mu\vec{b}$ : $d = \dfrac{|\vec{b} \times (\vec{a}_2 - \vec{a}_1)|}{|\vec{b}|}$ .
$d = 0$ for intersecting lines. For Cartesian lines, first read $\vec{a}_1$ , $\vec{b}_1$ , $\vec{a}_2$ , $\vec{b}_2$ .

The skew formula fails for parallel lines because $\vec{b}_1 \times \vec{b}_2 = \vec{0}$.
Lines	Shortest distance
Skew: $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ , $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$	$d = \dfrac{\|(\vec{b}_1 \times \vec{b}_2) \cdot (\vec{a}_2 - \vec{a}_1)\|}{\|\vec{b}_1 \times \vec{b}_2\|}$
Parallel: common direction $\vec{b}$	$d = \dfrac{\|\vec{b} \times (\vec{a}_2 - \vec{a}_1)\|}{\|\vec{b}\|}$
Intersecting	$d = 0$

Skew lines and their common perpendicular $MN$ (sketch, not to scale).

Cross, dot, divide.
Quantity	Value
$\vec{b}_1 \times \vec{b}_2$	$-3\hat{i} + 6\hat{j} + 6\hat{k}$
$\|\vec{b}_1 \times \vec{b}_2\|$	$9$
$\vec{a}_2 - \vec{a}_1$	$\hat{i} + 3\hat{k}$
Dot product	$-3 + 0 + 18 = 15$
$d$	$\dfrac{15}{9} = \dfrac53$

Parallel lines need their own formula.
Quantity	Value
$\vec{a}_2 - \vec{a}_1$	$2\hat{i} - 2\hat{j} + 3\hat{k}$
$\vec{b} \times (\vec{a}_2 - \vec{a}_1)$	$2\hat{i} - 7\hat{j} - 6\hat{k}$
Its length	$\sqrt{89}$
$\|\vec{b}\|$	$3$
$d$	$\dfrac{\sqrt{89}}{3}$

Formulas

Skew lines
$d = \dfrac{|(\vec{b}_1 \times \vec{b}_2) \cdot (\vec{a}_2 - \vec{a}_1)|}{|\vec{b}_1 \times \vec{b}_2|}$

Key terms

Shortest distance: The length of the common perpendicular of two lines.

Exam tips

How ISC marks are awarded

Direction cosines (2 marks): ratios and length, then cosines.
Angle between lines (2 marks): sum of products and lengths, then the angle.
Lines intersect (3 marks): general points, parameters, third equation checked and the point.
Foot and distance (3 marks): $\vec{PF}$ , $\lambda$ and foot, distance.
Shortest distance (3 marks): cross product and length, dot product, answer.

Common mistakes

Reading ratios from a Cartesian line that is not in standard form.
Forgetting the modulus for the acute angle.
Calling lines skew without first showing they are not parallel.
Using the skew formula for parallel lines (its denominator is $0$ ).
Not checking the third equation when finding a point of intersection.

1. Direction cosines and direction ratios

Formulas

Key terms

2. Coordinate axes and coordinate planes

Formulas

Key terms

3. Angle between two lines

Formulas

Key terms

4. Equations of a line

Formulas

Key terms

5. Coplanar and skew lines; intersection; distance from a line

Formulas

Key terms

6. Shortest distance

Formulas

Key terms

Exam tips

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