Mathematics · Textbook solutions

Three Dimensional Geometry

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 37 questions

11.2 Direction Cosines and Direction Ratios of a Line

10 q

Solved Examples

Worked · 5

11.1 Eg.1 $90^\circ$
11.1 Eg.2 $2$
11.1 Eg.3 $(-2, 4, -5)$
11.1 Eg.4 $x$
11.1 Eg.5 $A(2, 3, -4)$

Exercise 11.1

Practice · 5

Ex 11.1 Q1 $90^\circ$
Ex 11.1 Q2 $Find the direction cosines of a line which makes equal angles with the coordinate axes.$
Ex 11.1 Q3 $-18$
Ex 11.1 Q4 $(2, 3, 4)$
Ex 11.1 Q5 $(3, 5, -4)$

11.3 Equation of a Line in Space, 11.4 Angle between Two Lines and 11.5 Shortest Distance

22 q

Solved Examples

Worked · 5

11.2 Eg.6 $(5, 2, -4)$
11.2 Eg.7 $\vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$
11.2 Eg.8 $\frac{x + 3}{3} = \frac{y - 1}{5} = \frac{z + 3}{4}$
11.2 Eg.9 $l_1$
11.2 Eg.10 $l_1$

Exercise 11.2

Practice · 17

Ex 11.2 Q1 $\frac{12}{13},\ \frac{-3}{13},\ \frac{-4}{13}$
Ex 11.2 Q2 $(1, -1, 2)$
Ex 11.2 Q3 $(4, 7, 8)$
Ex 11.2 Q4 $(1, 2, 3)$
Ex 11.2 Q5 $2\hat{i} - \hat{j} + 4\hat{k}$
Ex 11.2 Q6 $(-2, 4, -5)$
Ex 11.2 Q7 $\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2}$
Find the angle between the following pairs of lines:
Ex 11.2 Q8(i) $\vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k})$
Ex 11.2 Q8(ii) $\vec{r} = 3\hat{i} + \hat{j} - 2\hat{k} + \lambda(\hat{i} - \hat{j} - 2\hat{k})$
Find the angle between the following pair of lines:
Ex 11.2 Q9(i) $\frac{x - 2}{2} = \frac{y - 1}{5} = \frac{z + 3}{-3}$
Ex 11.2 Q9(ii) $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$
Ex 11.2 Q10 $p$
Ex 11.2 Q11 $\frac{x - 5}{7} = \frac{y + 2}{-5} = \frac{z}{1}$
Ex 11.2 Q12 $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$
Ex 11.2 Q13 $\frac{x + 1}{7} = \frac{y + 1}{-6} = \frac{z + 1}{1}$
Ex 11.2 Q14 $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 3\hat{j} + 2\hat{k})$
Ex 11.2 Q15 $\vec{r} = (1 - t)\hat{i} + (t - 2)\hat{j} + (3 - 2t)\hat{k}$

Miscellaneous Exercise on Chapter 11

5 q

Miscellaneous Exercise

Practice · 5

Misc Q1 $a$
Misc Q2 $x$
Misc Q3 $\frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2}$
Misc Q4 $\vec{r} = 6\hat{i} + 2\hat{j} + 2\hat{k} + \lambda(\hat{i} - 2\hat{j} + 2\hat{k})$
Misc Q5 $(1, 2, -4)$

What the board has asked from this chapter

Questions per paper · 26 papers · 2022 to 2026

Bars average the full papers of each year. 44 further questions come from set variants and are listed below.

2026

31 q

1 $\dfrac{1 - x}{0} = y = z$
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Direction Cosines and Direction Ratios
2 $\frac{x}{0} = \frac{y}{0} = \frac{z}{1}$
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Equation of a Line in Space
3 $l_1$
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Direction Cosines and Direction Ratios
8 $x = y = 1 - z$
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Direction Cosines and Direction Ratios
9 $\frac{x}{0} = \frac{y}{1} = \frac{z}{0}$
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Equation of a Line in Space
10 $l_1$
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Direction Cosines and Direction Ratios
15 $\dfrac{2x - 1}{4} = \dfrac{1 - y}{3} = \dfrac{-z}{6}$
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Direction Cosines and Direction Ratios
16 $l_1, m_1, n_1$
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Angle Between Two Lines
16 $\frac{x}{1} = \frac{y}{0} = \frac{z}{0}$
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Equation of a Line in Space
17 $l_1$
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Direction Cosines and Direction Ratios
Direction : Question numbers 19 and 20 are Assertion (A) and Reason (R) based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
20 $< 1, 1, 1 >$
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Direction Cosines and Direction Ratios
Direction : Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
20 $x = py + q$
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Angle Between Two Lines
21 $x = \dfrac{y - 1}{2} = \dfrac{z - 2}{3}$
Equation of a Line in Space
23 $\dfrac{x}{1} = \dfrac{y + 1}{-1} = \dfrac{z - 3}{-2}$
Equation of a Line in Space
25 (a) $\dfrac{x - 3}{1} = \dfrac{1 - y}{1} = \dfrac{z + 2}{p}$
Angle Between Two Lines
25 (b) $\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k})$
Equation of a Line in Space
25 $\vec{r} = -\hat{j} + 3\hat{k} + \lambda(2\hat{i} - 2\hat{j} + \hat{k})$
Equation of a Line in Space
25 $\dfrac{x - 2}{3} = \dfrac{y + 5}{2} = \dfrac{1 - z}{-6}$
Angle Between Two Lines
30 (a) $\dfrac{x - 2}{3} = \dfrac{1 - y}{2} = \dfrac{z - 3}{2}$
Equation of a Line in Space
30 (b) $\vec{r} = (4 + \lambda)\hat{i} + (2\lambda - 1)\hat{j} - 3\lambda\hat{k}$
Shortest Distance Between Two Lines
32 $Show that line AB passing through points A(0, 4, 1), B(2, 3, -1) and the line CD passing through points C(4, 5, 0), D(2, 6, 2) are parallel. Also, find distance between them.$
Shortest Distance Between Two Lines
33 $Prove that the line through points A(0, -1, -1) and B(4, 5, 1) intersects the line through points C(3, 9, 4) and D(-4, 4, 4). Hence, write the equation of line passing through the point of intersection of lines AB and CD as well as origin.$
Equation of a Line in Space
35 (a) $\dfrac{-x - 3}{-5} = \dfrac{1 - y}{-2} = \dfrac{3z + 12}{9}$
Equation of a Line in Space
35 (a) $l_1$
Shortest Distance Between Two Lines
35 (b) $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$
Shortest Distance Between Two Lines
35 $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$
Equation of a Line in Space
35 $\vec{r} = 4\hat{i} + \hat{j} + \lambda(5\hat{i} + 2\hat{j} + \hat{k})$
Equation of a Line in Space
35 (b) $\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$
Shortest Distance Between Two Lines
35 $\dfrac{x - 1}{2} = \dfrac{y - 2}{3} = \dfrac{z - 3}{4}$
Equation of a Line in Space
35 $\dfrac{x - 6}{3} = \dfrac{y - 7}{2} = \dfrac{z - 7}{-2}$
Equation of a Line in Space
35 $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$
Shortest Distance Between Two Lines

2025

29 q

6 $\lambda, \lambda, \lambda$
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Direction Cosines and Direction Ratios
6 $\dfrac{3\pi}{4}, \dfrac{\pi}{3}$
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Direction Cosines and Direction Ratios
6 $(-2, 3, 5)$
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Direction Cosines and Direction Ratios
9 $3\hat{i} + \hat{j} + 2\hat{k}$
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Equation of a Line in Space
13 $x = 1 + 5\mu$
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Equation of a Line in Space
25 $\lambda$
Angle Between Two Lines
25 $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$
Shortest Distance Between Two Lines
25 $l_1 : \dfrac{x - 5}{2} = \dfrac{y + 3}{1} = \dfrac{z - 1}{-3}$
Angle Between Two Lines
29 (a) $\vec{r} = (1 - \lambda)\hat{i} + (\lambda - 2)\hat{j} + (3 - 2\lambda)\hat{k}$
Shortest Distance Between Two Lines
30 (a) $P(2,\ 4,\ -1)$
Equation of a Line in Space
30 (b) $3\hat{i} - \hat{j} - 2\hat{k}$
Equation of a Line in Space
30 $(-1, -5, -10)$
Equation of a Line in Space
31 (b) $\vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda(\hat{i} - 2\hat{j} + 3\hat{k})$
Shortest Distance Between Two Lines
34 (a) $\dfrac{x + 1}{2} = \dfrac{y - 1}{1} = \dfrac{z - 9}{-3}$
Shortest Distance Between Two Lines
34 (a) $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$
Equation of a Line in Space
34 (b) $l : \vec{r} = 4\hat{i} + 2\hat{j} + 2\hat{k} + \lambda\,(\hat{i} - \hat{j} - \hat{k})$
Equation of a Line in Space
34 (b) $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}$
Equation of a Line in Space
35 (a) $(1, 1, 4)$
Equation of a Line in Space
35 (a) $\dfrac{2x + 4}{6} = \dfrac{y + 1}{2} = \dfrac{-2z + 6}{-4}$
Equation of a Line in Space
35 (b) $\dfrac{x-1}{3} = \dfrac{y+1}{2} = \dfrac{z-4}{3}$
Equation of a Line in Space
35 (b) $(1, 2, -4)$
Equation of a Line in Space
35 $A(0,\ -1,\ -1)$
Shortest Distance Between Two Lines
35 $(2, -1, 5)$
Equation of a Line in Space
35 (b) $(-1, 5, 2)$
Equation of a Line in Space
35 $\dfrac{x}{1} = \dfrac{1 - y}{-2} = \dfrac{2z - 4}{6}$
Equation of a Line in Space
An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1 : \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 3}{4}$ , while the track for Line B is represented by $l_2 : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z + 2}{-3}$ . Based on the above information, answer the following questions :
37 (i) $Find whether the two metro tracks are parallel.$
Angle Between Two Lines
An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1 : \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 3}{4}$ , while the track for Line B is represented by $l_2 : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z + 2}{-3}$ . Based on the above information, answer the following questions :
37 (ii) $(l_1)$
Equation of a Line in Space
An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1 : \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 3}{4}$ , while the track for Line B is represented by $l_2 : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z + 2}{-3}$ . Based on the above information, answer the following questions :
37 (iii) (a) $To connect the stations, a pedestrian pathway perpendicular to the two metro lines is to be constructed which passes through point (3, 2, 1). Determine the equation of the pedestrian walkway.$
Equation of a Line in Space
An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $l_1 : \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 3}{4}$ , while the track for Line B is represented by $l_2 : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z + 2}{-3}$ . Based on the above information, answer the following questions :
37 (iii) (b) $Find the shortest distance between Line A and Line B.$
Shortest Distance Between Two Lines

2024

27 q

9 $The coordinates of the foot of the perpendicular drawn from the point (0, 1, 2) on the x-axis are given by :$
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Equation of a Line in Space
9 $\alpha$
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Direction Cosines and Direction Ratios
9 $\sqrt{3}\,k$
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Direction Cosines and Direction Ratios
14 $\vec{a} = \hat{i} - \hat{j}$
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Equation of a Line in Space
14 $\vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda(3\hat{i} - \hat{j})$
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Equation of a Line in Space
14 $(1, -1, 0)$
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Equation of a Line in Space
15 $\dfrac{1-x}{2} = \dfrac{y-1}{3} = \dfrac{z}{1}$
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Angle Between Two Lines
16 $The distance of point P(a, b, c) from y-axis is :$
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Direction Cosines and Direction Ratios
16 $\dfrac{x}{1} = \dfrac{y}{-1} = \dfrac{z}{0}$
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Direction Cosines and Direction Ratios
16 $\dfrac{x-1}{2} = -y = \dfrac{2z+1}{6}$
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Direction Cosines and Direction Ratios
16 $\dfrac{\pi}{4}$
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Direction Cosines and Direction Ratios
17 $(1, -3, 2)$
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Equation of a Line in Space
18 $30^\circ$
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Direction Cosines and Direction Ratios
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
20 $\alpha$
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Direction Cosines and Direction Ratios
25 $(2, 3, -5)$
Equation of a Line in Space
32 $\vec{r} = \lambda\hat{i} + (2\lambda + 1)\hat{j} + (3\lambda + 2)\hat{k}$
Equation of a Line in Space
32 $l_2$
Equation of a Line in Space
33 $\dfrac{x-8}{3} = \dfrac{y+19}{-16} = \dfrac{z-10}{7}$
Equation of a Line in Space
34 $\Delta$
Equation of a Line in Space
34 $\frac{x + 1}{1} = \frac{y - 2}{-2} = \frac{z - 1}{2}$
Equation of a Line in Space
34 (a) $(2, 3, -8)$
Equation of a Line in Space
35 (a) $\dfrac{x}{1} = \dfrac{y - 1}{2} = \dfrac{z - 2}{3}$
Equation of a Line in Space
34 (b) $\mathrm{L}_1$
Shortest Distance Between Two Lines
35 (b) $A(-1, 2, 1)$
Shortest Distance Between Two Lines
35 $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$
Equation of a Line in Space
35 (a) $\dfrac{x}{2} = \dfrac{2y-6}{4} = \dfrac{1-z}{-1}$
Shortest Distance Between Two Lines
35 (b) $\dfrac{x-1}{-3} = \dfrac{y-2}{2k} = \dfrac{z-3}{2}$
Angle Between Two Lines

2023

40 q

2 $The direction ratios of a line parallel to z-axis are :$
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Direction Cosines and Direction Ratios
4 $(2, -1, 0)$
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Equation of a Line in Space
7 $^\circ$
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Equation of a Line in Space
7 $Direction cosines of a line perpendicular to both x-axis and z-axis are :$
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Direction Cosines and Direction Ratios
9 $\frac{x+1}{2} = \frac{y+2}{3} = \frac{z+3}{4}$
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Equation of a Line in Space
11 $\frac{x - 1}{2} = \frac{1 - y}{3} = \frac{2z - 1}{12}$
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Direction Cosines and Direction Ratios
11 $Equation of a line passing through point (1, 1, 1) and parallel to z-axis is$
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Equation of a Line in Space
13 $\lambda$
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Angle Between Two Lines
14 $Distance of the point (p, q, r) from y-axis is :$
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Direction Cosines and Direction Ratios
15 $90^{\circ}$
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Direction Cosines and Direction Ratios
16 $2x = 3y = -z$
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Angle Between Two Lines
17 $\left(\dfrac{1}{a}, \dfrac{1}{a}, \dfrac{1}{a}\right)$
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Direction Cosines and Direction Ratios
18 $Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is$
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Equation of a Line in Space
Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (a), (b), (c) and (d) as given below.
20 $(3, -1, 3)$
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Equation of a Line in Space
Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (a), (b), (c) and (d) as given below.
20 $(-1, -2, 1)$
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Angle Between Two Lines
In the following questions 19 & 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices :
20 $\alpha, \beta, \gamma$
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Direction Cosines and Direction Ratios
Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (a), (b), (c) and (d) as given below.
20 $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$
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Angle Between Two Lines
22 $\frac{x-5}{\alpha} = \frac{y+2}{-5} = \frac{z + \frac{24}{5}}{\beta}$
Angle Between Two Lines
23 $x = ay + b$
Direction Cosines and Direction Ratios
23 $-$
Direction Cosines and Direction Ratios
24 $\frac{x}{1} = \frac{y - 1}{2} = \frac{z + 1}{2}$
Equation of a Line in Space
23 (b) $x = 2\lambda + 2$
Angle Between Two Lines
25 (a) $\dfrac{x-1}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{3}$
Equation of a Line in Space
25 $\frac{x-1}{-2} = \frac{y-4}{3p} = \frac{z-3}{4}$
Angle Between Two Lines
25 $-1$
Equation of a Line in Space
25 $\vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k})$
Angle Between Two Lines
25 (b) $5x - 3 = 15y + 7 = 3 - 10z$
Direction Cosines and Direction Ratios
28 (a) $\dfrac{x - 15}{3} = \dfrac{y - 29}{8} = \dfrac{z - 5}{-5}$
Equation of a Line in Space
30 (a) $\dfrac{x + 3}{5} = \dfrac{y - 1}{2} = \dfrac{z + 4}{3}$
Equation of a Line in Space
31 $\vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$
Shortest Distance Between Two Lines
32 (a) $(2, -1, 5)$
Equation of a Line in Space
32 (b) $\Delta$
Equation of a Line in Space
33 (a) $-6$
Equation of a Line in Space
34 (a) $(1, 2, -4)$
Shortest Distance Between Two Lines
33 (b) $l$
Equation of a Line in Space
35 (a) $\frac{x-1}{2} = \frac{y-b}{3} = \frac{z-3}{4}$
Equation of a Line in Space
34 (b) $Find the equations of the line passing through the points A(1, 2, 3) and B(3, 5, 9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.$
Equation of a Line in Space
35 (a) $\dfrac{x-1}{3} = \dfrac{y+1}{2} = \dfrac{z-1}{5}$
Shortest Distance Between Two Lines
35 (b) $-1, -2, 1$
Equation of a Line in Space
35 (b) $2x = 3y = -z$
Angle Between Two Lines

2022

26 q

1 $2\hat{i} - \hat{j} + \hat{k}$
Equation of a Line in Space
1 $\dfrac{2x - 1}{12} = \dfrac{y + 2}{2} = \dfrac{z - 3}{3}$
Direction Cosines and Direction Ratios
3 $(-2, -1, -3)$
The Plane
4 $x - y + z + \lambda = 0$
The Plane
5 $\alpha$
Direction Cosines and Direction Ratios
6 $\lambda$
The Plane
6 $P(2, 2, 1)$
Equation of a Line in Space
7 $\vec{r} = 3\hat{i} + 5\hat{j} + 7\hat{k} + \lambda(\hat{i} - 2\hat{j} + \hat{k})$
Shortest Distance Between Two Lines
8 $\frac{x - 3}{-1} = \frac{y + 4}{1} = \frac{z + 5}{6}$
The Plane
8 $\vec{r} \cdot (2\hat{i} + 2\hat{j} - 3\hat{k}) = 7$
The Plane
9 $\vec{r} \cdot \left(\hat{i} + \hat{j} + \hat{k}\right) = 10$
The Plane
10 (a) $60^\circ$
Direction Cosines and Direction Ratios
10 (a) $(2, 1, 0)$
The Plane
10 $\frac{x + 2}{3} = \frac{y + 1}{2} = \frac{z - 3}{2}$
Equation of a Line in Space
10 (b) $x = \frac{y - 1}{2} = \frac{z - 2}{3}$
Shortest Distance Between Two Lines
10 $(2, 3, 4)$
The Plane
10 $\dfrac{1 - x}{2} = \dfrac{y - 3}{4} = \dfrac{z}{-1}$
Shortest Distance Between Two Lines
10 (b) $\dfrac{x - 1}{2} = \dfrac{y - 2}{3} = \dfrac{z - 3}{4}$
Shortest Distance Between Two Lines
11 $(-1, -5, -10)$
The Plane
11 $\vec{r} \cdot (\hat{i} + 3\hat{j}) = 6$
The Plane
13 $\vec{r} = 4\hat{i} + 2\hat{j} + 7\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k})$
The Plane
13 $(5, 1, 6)$
Equation of a Line in Space
Electrical transmission wires which are laid down in winters are stretched tightly to accommodate expansion in summers. Two such wires lie along the following lines : $l_1 : \frac{x + 1}{3} = \frac{y - 3}{-2} = \frac{z + 2}{-1}$ $l_2 : \frac{x}{-1} = \frac{y - 7}{3} = \frac{z + 7}{-2}$ Based on the given information, answer the following questions :
14 (i) $l_1$
Shortest Distance Between Two Lines
Two motorcycles A and B are running at the speed more than the allowed speed on the roads represented by the lines $\vec{r} = \lambda\left(\hat{i} + 2\hat{j} - \hat{k}\right)$ and $\vec{r} = \left(3\hat{i} + 3\hat{j}\right) + \mu\left(2\hat{i} + \hat{j} + \hat{k}\right)$ respectively. Based on the above information, answer the following questions :
14 (a) $Find the shortest distance between the given lines.$
Shortest Distance Between Two Lines
Two motorcycles A and B are running at the speed more than the allowed speed on the roads represented by the lines $\vec{r} = \lambda\left(\hat{i} + 2\hat{j} - \hat{k}\right)$ and $\vec{r} = \left(3\hat{i} + 3\hat{j}\right) + \mu\left(2\hat{i} + \hat{j} + \hat{k}\right)$ respectively. Based on the above information, answer the following questions :
14 (b) $Find the point at which the motorcycles may collide.$
Equation of a Line in Space
Electrical transmission wires which are laid down in winters are stretched tightly to accommodate expansion in summers. Two such wires lie along the following lines : $l_1 : \frac{x + 1}{3} = \frac{y - 3}{-2} = \frac{z + 2}{-1}$ $l_2 : \frac{x}{-1} = \frac{y - 7}{3} = \frac{z + 7}{-2}$ Based on the given information, answer the following questions :
14 (ii) $l_1$
Equation of a Line in Space

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 4 times: 2022*, 2023*, 2025*, 2026 (* = numbers or wording changed that year)
35 $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$
Question 2
Asked 3 times: 2023*, 2024*, 2025 (* = numbers or wording changed that year)
6 $\lambda, \lambda, \lambda$
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Question 3
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
35 (a) $\dfrac{-x - 3}{-5} = \dfrac{1 - y}{-2} = \dfrac{3z + 12}{9}$
Question 4
Asked 2 times: 2025, 2026
33 $Prove that the line through points A(0, -1, -1) and B(4, 5, 1) intersects the line through points C(3, 9, 4) and D(-4, 4, 4). Hence, write the equation of line passing through the point of intersection of lines AB and CD as well as origin.$
Question 5
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
35 $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$
Question 6
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
35 (b) $\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$
Question 7
Asked 2 times: 2024*, 2025 (* = numbers or wording changed that year)
34 (a) $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$
Question 8
Asked 2 times: 2023*, 2025 (* = numbers or wording changed that year)
35 $(2, -1, 5)$
Question 9
Asked 2 times: 2024*, 2025 (* = numbers or wording changed that year)
35 (b) $(1, 2, -4)$
Question 10
Asked 2 times: 2023, 2024
16 $The distance of point P(a, b, c) from y-axis is :$
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11.2 Direction Cosines and Direction Ratios of a Line

Solved Examples

Exercise 11.1

11.3 Equation of a Line in Space, 11.4 Angle between Two Lines and 11.5 Shortest Distance

Solved Examples

Exercise 11.2

Miscellaneous Exercise on Chapter 11

Miscellaneous Exercise

What the board has asked from this chapter

2026

2025

2024

2023

2022

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10