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CBSE Class 12 · Chapter 11 · Vectors and Three-Dimensional Geometry · 2026-27

Three Dimensional Geometry Class 12: MCQ and case study questions

15 multiple-choice questions and 2 case-based questions on the current (rationalised) syllabus, in the style of the board paper's Sections A and E. Try each one first; the answer, a one-line reason and a worked solution open on a tap.

  • 15 MCQs (1 mark each)
  • 2 case studies (4 marks each)
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Multiple-choice questions

Choose one option. Section A of the board paper has 18 MCQs of 1 mark each, spread over all chapters.

Q1

·1 mark·Multiple choiceDirection cosines and ratios

A line passes through \(P(2, 3, -1)\) and \(Q(4, 4, 1)\). Which triple gives the direction cosines of \(\overrightarrow{PQ}\)?

  1. (a)\(\dfrac23, \dfrac13, \dfrac23\)
  2. (b)\(\dfrac13, \dfrac23, \dfrac23\)
  3. (c)\(\dfrac29, \dfrac19, \dfrac29\)
  4. (d)\(\dfrac{2}{\sqrt5}, \dfrac{1}{\sqrt5}, \dfrac{2}{\sqrt5}\)
Show answer
Answer: (a) \(\dfrac23, \dfrac13, \dfrac23\)

Why: Direction ratios 2, 1, 2 have magnitude 3.

\(\overrightarrow{PQ} = 2\hat{i} + \hat{j} + 2\hat{k}\), \(|\overrightarrow{PQ}| = \sqrt{4 + 1 + 4} = 3\). Direction cosines \(\dfrac23, \dfrac13, \dfrac23\).

Q2

·1 mark·Multiple choiceDirection cosines and ratios

If \(l, m, n\) are the direction cosines of a line, then \(l^2 + m^2 + n^2\) equals

  1. (a)\(0\)
  2. (b)\(1\)
  3. (c)\(3\)
  4. (d)\(l + m + n\)
Show answer
Answer: (b) \(1\)

Why: cos²α + cos²β + cos²γ = 1.

\(l, m, n\) are the components of a unit vector along the line, so \(l^2 + m^2 + n^2 = 1\).

Q3

·1 mark·Multiple choiceEquation of a line

The vector equation of the line through \((1, -1, 2)\) parallel to \(3\hat{i} + \hat{j} - \hat{k}\) is

  1. (a)\(\vec r = (3\hat{i} + \hat{j} - \hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k})\)
  2. (b)\(\vec r = \lambda(\hat{i} - \hat{j} + 2\hat{k})\)
  3. (c)\(\vec r = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(3\hat{i} + \hat{j} - \hat{k})\)
  4. (d)\(\vec r = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k})\)
Show answer
Answer: (c) \(\vec r = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(3\hat{i} + \hat{j} - \hat{k})\)

Why: r = (position vector of the point) + λ(direction vector).

\(\vec r = \vec a + \lambda\vec b\) with \(\vec a = \hat{i} - \hat{j} + 2\hat{k}\) (a point on the line) and \(\vec b = 3\hat{i} + \hat{j} - \hat{k}\) (its direction).

Q4

·1 mark·Multiple choiceEquation of a line

The Cartesian equations of the line through \((2, 0, -3)\) with direction ratios \(1, -2, 4\) are

  1. (a)\(\dfrac{x + 2}{1} = \dfrac{y}{-2} = \dfrac{z - 3}{4}\)
  2. (b)\(\dfrac{x - 1}{2} = \dfrac{y + 2}{0} = \dfrac{z - 4}{-3}\)
  3. (c)\(\dfrac{x - 2}{1} = \dfrac{y}{2} = \dfrac{z + 3}{4}\)
  4. (d)\(\dfrac{x - 2}{1} = \dfrac{y}{-2} = \dfrac{z + 3}{4}\)
Show answer
Answer: (d) \(\dfrac{x - 2}{1} = \dfrac{y}{-2} = \dfrac{z + 3}{4}\)

Why: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.

\(\dfrac{x - 2}{1} = \dfrac{y - 0}{-2} = \dfrac{z - (-3)}{4}\), i.e. \(\dfrac{x - 2}{1} = \dfrac{y}{-2} = \dfrac{z + 3}{4}\).

Q5

·1 mark·Multiple choiceAngle between two lines

The angle between two lines with direction ratios \(1, 2, 2\) and \(2, -2, 1\) is

  1. (a)\(\dfrac{\pi}{2}\)
  2. (b)\(\dfrac{\pi}{3}\)
  3. (c)\(\dfrac{\pi}{4}\)
  4. (d)\(0\)
Show answer
Answer: (a) \(\dfrac{\pi}{2}\)

Why: 1·2 + 2·(−2) + 2·1 = 0.

\(\cos\theta = \dfrac{|2 - 4 + 2|}{3 \cdot 3} = 0 \Rightarrow \theta = \dfrac{\pi}{2}\).

Q6

·1 mark·Multiple choiceAngle between two lines

The angle between two lines with direction ratios \(1, 0, 1\) and \(0, 1, 1\) is

  1. (a)\(\dfrac{\pi}{6}\)
  2. (b)\(\dfrac{\pi}{3}\)
  3. (c)\(\dfrac{\pi}{2}\)
  4. (d)\(\dfrac{\pi}{4}\)
Show answer
Answer: (b) \(\dfrac{\pi}{3}\)

Why: cos θ = 1/(√2 · √2) = 1/2.

\(\cos\theta = \dfrac{0 + 0 + 1}{\sqrt2\sqrt2} = \dfrac12 \Rightarrow \theta = \dfrac{\pi}{3}\).

Q7

·1 mark·Multiple choiceAngle between two lines

The lines \(\dfrac{x - 1}{2} = \dfrac{y}{k} = \dfrac{z}{3}\) and \(\dfrac{x}{1} = \dfrac{y}{2} = \dfrac{z}{-2}\) are perpendicular when \(k\) equals

  1. (a)\(-2\)
  2. (b)\(4\)
  3. (c)\(2\)
  4. (d)\(1\)
Show answer
Answer: (c) \(2\)

Why: a₁a₂ + b₁b₂ + c₁c₂ = 2 + 2k − 6 = 0.

\(2(1) + k(2) + 3(-2) = 0 \Rightarrow 2k - 4 = 0 \Rightarrow k = 2\).

Q8

·1 mark·Multiple choiceAngle between two lines

The lines with direction ratios \(2, -1, 4\) and \(-4, 2, \mu\) are parallel when \(\mu\) equals

  1. (a)\(8\)
  2. (b)\(-2\)
  3. (c)\(2\)
  4. (d)\(-8\)
Show answer
Answer: (d) \(-8\)

Why: Parallel lines have proportional direction ratios: −4/2 = 2/(−1) = μ/4.

\(\dfrac{-4}{2} = \dfrac{2}{-1} = -2\), so \(\dfrac{\mu}{4} = -2 \Rightarrow \mu = -8\).

Q9

·1 mark·Multiple choiceEquation of a line

Which of the following points lies on the line \(\dfrac{x - 2}{1} = \dfrac{y + 1}{2} = \dfrac{z - 3}{-2}\)?

  1. (a)\((4, 3, -1)\)
  2. (b)\((3, 1, 2)\)
  3. (c)\((2, 1, 3)\)
  4. (d)\((0, -5, 6)\)
Show answer
Answer: (a) \((4, 3, -1)\)

Why: General point (2 + λ, −1 + 2λ, 3 − 2λ); λ = 2 gives (4, 3, −1).

A general point is \((2 + \lambda, -1 + 2\lambda, 3 - 2\lambda)\). \(\lambda = 2\) gives \((4, 3, -1)\). For the others the three ratios differ (e.g. \((3, 1, 2)\): \(1, 1, \tfrac12\)).

Q10

·1 mark·Multiple choiceShortest distance between two lines

The shortest distance between the \(x\)-axis and the line \(\vec r = (2\hat{j} + 3\hat{k}) + \mu\hat{k}\) is

  1. (a)\(0\)
  2. (b)\(2\)
  3. (c)\(3\)
  4. (d)\(\sqrt{13}\)
Show answer
Answer: (b) \(2\)

Why: The x-axis is r = λî; b₁ × b₂ = î × k̂ = −ĵ.

\(\vec b_1 \times \vec b_2 = \hat i \times \hat k = -\hat j\); \(\vec a_2 - \vec a_1 = 2\hat j + 3\hat k\). SD \(= |(2\hat j + 3\hat k)\cdot(-\hat j)| = 2\). (The second line is the vertical line \(x = 0\), \(y = 2\).)

Q11

·1 mark·Multiple choiceShortest distance between two lines

The distance between the parallel lines \(\vec r = (\hat{i} + 2\hat{j}) + \lambda(\hat{i} - \hat{j} + \hat{k})\) and \(\vec r = (2\hat{i} + 2\hat{j}) + \mu(2\hat{i} - 2\hat{j} + 2\hat{k})\) is

  1. (a)\(\dfrac{\sqrt6}{3}\)
  2. (b)\(\sqrt2\)
  3. (c)\(\dfrac{1}{\sqrt3}\)
  4. (d)\(0\)
Show answer
Answer: (a) \(\dfrac{\sqrt6}{3}\)

Why: For parallel lines, d = |(a₂ − a₁) × b|/|b|.

\(\vec b = \hat{i} - \hat{j} + \hat{k}\), \(\vec a_2 - \vec a_1 = \hat i\). \(\hat i \times \vec b = -\hat j - \hat k\), magnitude \(\sqrt2\). \(d = \dfrac{\sqrt2}{\sqrt3} = \dfrac{\sqrt6}{3}\).

Q12

·1 mark·Multiple choiceDirection cosines and ratios

The direction ratios of the line \(\dfrac{x - 1}{3} = \dfrac{2 - y}{4} = \dfrac{z}{5}\) are

  1. (a)\(3, 4, 5\)
  2. (b)\(3, -4, 5\)
  3. (c)\(1, 2, 0\)
  4. (d)\(-3, 4, 5\)
Show answer
Answer: (b) \(3, -4, 5\)

Why: Rewrite (2 − y)/4 as (y − 2)/(−4).

\(\dfrac{2 - y}{4} = \dfrac{y - 2}{-4}\), so the line is \(\dfrac{x - 1}{3} = \dfrac{y - 2}{-4} = \dfrac{z - 0}{5}\): direction ratios \(3, -4, 5\).

Q13

·1 mark·Multiple choiceDirection cosines and ratios

A line makes angles of \(90^\circ\), \(60^\circ\) and \(30^\circ\) with the positive \(x\)-, \(y\)- and \(z\)-axes. Its direction cosines are

  1. (a)\(0, \dfrac{\sqrt3}{2}, \dfrac12\)
  2. (b)\(1, \dfrac12, \dfrac{\sqrt3}{2}\)
  3. (c)\(0, \dfrac12, \dfrac{\sqrt3}{2}\)
  4. (d)\(0, \dfrac12, \dfrac12\)
Show answer
Answer: (c) \(0, \dfrac12, \dfrac{\sqrt3}{2}\)

Why: cos 90° = 0, cos 60° = 1/2, cos 30° = √3/2.

\((\cos 90^\circ, \cos 60^\circ, \cos 30^\circ) = \left(0, \dfrac12, \dfrac{\sqrt3}{2}\right)\); check: \(0 + \tfrac14 + \tfrac34 = 1\).

Q14

·1 mark·Multiple choiceDirection cosines and ratios

A line makes an acute angle \(\alpha\) with the \(x\)-axis, \(45^\circ\) with the \(y\)-axis and \(60^\circ\) with the \(z\)-axis. Then \(\cos\alpha\) is

  1. (a)\(\dfrac14\)
  2. (b)\(\dfrac{\sqrt3}{2}\)
  3. (c)\(\dfrac12\)
  4. (d)\(\dfrac{1}{\sqrt2}\)
Show answer
Answer: (c) \(\dfrac12\)

Why: cos²α + cos²45° + cos²60° = 1 gives cos²α = 1/4.

\(\cos^2\alpha + \dfrac12 + \dfrac14 = 1 \Rightarrow \cos^2\alpha = \dfrac14\); \(\alpha\) is acute, so \(\cos\alpha = \dfrac12\) (\(\alpha = 60^\circ\)). (\(\tfrac14\) is \(\cos^2\alpha\), not \(\cos\alpha\).)

Q15

·1 mark·Multiple choiceShortest distance between two lines

The shortest distance between the lines \(\vec r = \lambda(\hat{i} + \hat{j})\) and \(\vec r = 3\hat{k} + \mu(\hat{i} - \hat{j})\) is

  1. (a)\(0\)
  2. (b)\(\dfrac32\)
  3. (c)\(6\)
  4. (d)\(3\)
Show answer
Answer: (d) \(3\)

Why: b₁ × b₂ = −2k; (a₂ − a₁)·(b₁ × b₂) = 3k·(−2k) = −6.

\(\vec b_1 \times \vec b_2 = (\hat{i} + \hat{j}) \times (\hat{i} - \hat{j}) = -2\hat k\). SD \(= \dfrac{|3\hat k \cdot (-2\hat k)|}{2} = \dfrac{6}{2} = 3\).

Case study questions

Section E of the board paper has three case-based questions of 4 marks: a real-life passage, then parts of 1, 1 and 2 marks, with a choice (OR) on the 2-mark part.

Case study 1: Zip-line cable (4 marks)

An adventure park stretches a straight zip-line cable from the top of a tower at \(A(2, 1, 6)\) to a landing post at \(B(6, 3, 2)\), with coordinates in metres and the \(z\)-axis vertical.

(i) Find the direction cosines of \(\overrightarrow{AB}\). [1 mark]
Show answer
Answer: \(\dfrac23, \dfrac13, -\dfrac23\)
Direction ratios \(4, 2, -4\), magnitude \(6\): direction cosines \(\dfrac23, \dfrac13, -\dfrac23\). A1
(ii) Find the length of the cable. [1 mark]
Show answer
Answer: \(6\) m
\(|AB| = \sqrt{16 + 4 + 16} = 6\) m. A1
(iii) Write a vector equation of the cable's line and find the point of the cable at height \(4\) m. [2 marks]
Show answer
Answer: \(\vec r = (2\hat{i} + \hat{j} + 6\hat{k}) + \lambda(2\hat{i} + \hat{j} - 2\hat{k})\); \((4, 2, 4)\)
\(\vec r = (2\hat{i} + \hat{j} + 6\hat{k}) + \lambda(2\hat{i} + \hat{j} - 2\hat{k})\) M1; \(6 - 2\lambda = 4 \Rightarrow \lambda = 1\), point \((4, 2, 4)\). A1
OR Show that if the cable were extended beyond \(B\), it would reach the ground (\(z = 0\)) at \((8, 4, 0)\). [2 marks]
Show answer
Answer: \(\lambda = 3\) gives \((8, 4, 0)\)
On \(\vec r = (2\hat{i} + \hat{j} + 6\hat{k}) + \lambda(2\hat{i} + \hat{j} - 2\hat{k})\), \(z = 0 \Rightarrow \lambda = 3\) M1, giving \((2 + 6, 1 + 3, 0) = (8, 4, 0)\). A1

Case study 2: Two drone flight paths (4 marks)

Two delivery drones fly along straight paths. Drone \(P\) follows \(\vec r = (\hat{i} + 2\hat{j}) + s(\hat{i} + \hat{k})\) and drone \(Q\) follows \(\vec r = 3\hat{k} + t(\hat{j} + \hat{k})\) (units: hundreds of metres). Air-traffic rules require the paths to stay well apart.

(i) Find the angle between the two paths. [1 mark]
Show answer
Answer: \(\dfrac{\pi}{3}\)
\(\cos\theta = \dfrac{|0 + 0 + 1|}{\sqrt2\sqrt2} = \dfrac12 \Rightarrow \theta = \dfrac{\pi}{3}\). A1
(ii) Find \(\vec b_1 \times \vec b_2\) for the two direction vectors. [1 mark]
Show answer
Answer: \(-\hat{i} - \hat{j} + \hat{k}\)
\((\hat{i} + \hat{k}) \times (\hat{j} + \hat{k}) = -\hat{i} - \hat{j} + \hat{k}\). A1
(iii) Find the shortest distance between the two paths. [2 marks]
Show answer
Answer: \(2\sqrt3\)
\(\vec a_2 - \vec a_1 = -\hat{i} - 2\hat{j} + 3\hat{k}\); \((\vec a_2 - \vec a_1)\cdot(\vec b_1 \times \vec b_2) = 1 + 2 + 3 = 6\) M1; SD \(= \dfrac{6}{\sqrt3} = 2\sqrt3\) (hundred metres). A1
OR Show that the two paths do not meet. [2 marks]
Show answer
Answer: The equations are inconsistent
Equate points: \(1 + s = 0\), \(2 = t\), \(s = 3 + t\) M1. The first two give \(s = -1\), \(t = 2\), but then \(s = 3 + t = 5 \ne -1\): no common point. A1

Next steps for Three Dimensional Geometry

This free set is separate from the chapter's question bank. On the Three Dimensional Geometry chapter page the revision notes and the first step of the Route to 95 are free for everyone. CBSE Essentials adds all 40 questions in the chapter bank (short and long answers, assertion–reason and more case studies) with their full step-marking schemes, the Route to 95 checkpoints with your progress saved, Skill Builders and the full common-mistakes library. There is no AI marking on CBSE Math Revision: you check your work against the marking scheme.

Three Dimensional Geometry: notes and questionsNCERT solutions, Chapter 11What Essentials includes

Original questions written by CBSE Math Revision for the CBSE 2026-27 syllabus; not taken from NCERT, NCERT Exemplar or CBSE papers. Every answer was re-checked by computer algebra and by an independent reviewer. CBSE Math Revision is independent and not affiliated with CBSE or NCERT. Spotted a slip? Tell us and it goes in the corrections log.