Class 12 · Chapter 11 · Vectors and Three-Dimensional Geometry unit (14 of 80 marks)

Three Dimensional Geometry Class 12: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 46 board-style questions.

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46 questions
12 multiple choice, 3 assertion–reason, 11 very short answer, 11 short answer, 6 long answer, 3 case study
About 7 hours to master

Unit IV Vectors and Three-Dimensional Geometry: 14 marks of the 80-mark paper (shared with Chapter 10) — CBSE Curriculum 2025-26 Mathematics (041) (official PDF).

NCERT textbook: Chapter 11 on ncert.nic.in (PDF)

Revision notes

Three Dimensional Geometry — revision notes

1. Direction cosines and ratios

If a line makes angles $α, β, γ$ with the axes, $l = \cos α, m = \cos β, n = \cos γ$ and $l^{2} + m^{2} + n^{2} = 1$ .
Direction ratios $a, b, c$ are proportional to $l, m, n$ : $l = \frac{\pm a}{\sqrt{a^{2} + b^{2} + c^{2}}}$ , etc. For $P Q$ : $x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}$ .

2. Equation of a line

Through $\vec{a}$ , parallel to $\vec{b}$ : $\vec{r} = \vec{a} + λ \vec{b}$ ; Cartesian $\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}$ .
Through two points: $\vec{r} = \vec{a} + λ (\vec{b} - \vec{a})$ .
Standard form needs coefficient $1$ for $x, y, z$ : $\frac{3 - x}{2} = \frac{x - 3}{- 2}$ , $\frac{2 z - 1}{6} = \frac{z - \frac{1}{2}}{3}$ .
General point $(x_{1} + a λ, y_{1} + b λ, z_{1} + c λ)$ — use it for points on the line, intersections and feet of perpendiculars.

3. Angle between two lines

$\cos θ = \frac{| a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} |}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}}$ . Perpendicular: $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$ ; parallel: $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ .

4. Shortest distance

Skew lines (neither parallel nor intersecting): $d = \frac{| ({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2}) |}{| {\vec{b}}_{1} \times {\vec{b}}_{2} |}$ . $d = 0$ (non-parallel) $⟺$ the lines intersect.
Parallel lines $\vec{r} = {\vec{a}}_{i} + λ \vec{b}$ : $d = \frac{| \vec{b} \times ({\vec{a}}_{2} - {\vec{a}}_{1}) |}{| \vec{b} |}$ .
Foot of perpendicular from $P$ to a line: take the general point $F$ , solve $\vec{P F} \cdot \vec{b} = 0$ . Image $P' = 2 F - P$ ; distance $= P F$ .

Worked example 1

Line through $(1, 0, - 1)$ and $(3, 2, 0)$ : $\frac{x - 1}{2} = \frac{y}{2} = \frac{z + 1}{1}$ ; direction cosines $\frac{2}{3}, \frac{2}{3}, \frac{1}{3}$ .

Worked example 2

$\vec{r} = λ \hat{k}$ and $\vec{r} = \hat{i} + μ \hat{j}$ : ${\vec{b}}_{1} \times {\vec{b}}_{2} = \hat{k} \times \hat{j} = - \hat{i}$ , $({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot (- \hat{i}) = - 1$ , so $d = 1$ .

Worked example 3

Foot from $P (0, 0, 3)$ to $\vec{r} = λ (\hat{i} + \hat{j} + \hat{k})$ : $(λ, λ, λ - 3) \cdot (1, 1, 1) = 3 λ - 3 = 0$ , foot $(1, 1, 1)$ , distance $\sqrt{1 + 1 + 4} = \sqrt{6}$ .

Common errors

Reading direction ratios before putting the line in standard form (signs and coefficients of $x, y, z$ ).
Omitting the modulus in the angle formula (the angle between lines is taken acute) or in the distance formula.
Using the skew-line formula for parallel lines (it gives $0 / 0$ ).
Declaring lines intersecting after matching only two coordinates — always check the third.

Board-exam tips

The 5-mark question is usually shortest distance, intersection, or foot/image of a point: learn the general-point method thoroughly.
Planes (and angles/distances involving planes) are not in the current syllabus.

Topics in this chapter: Direction cosines and direction ratios of a line · Equation of a line (vector and Cartesian) · Angle between two lines · Skew lines and shortest distance · Perpendicular from a point to a line.

Formulas for this chapter: Chapter 11 on the Class 12 formula sheet (free PDF and A3 poster).

Three Dimensional Geometry Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 216 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceDirection cosines and ratios

What are the direction cosines of the line from $A (1, 0, 2)$ to $B (3, 3, 8)$ ?

Show worked answer

Direction ratios $2, 3, 6$ ; length $\sqrt{4 + 9 + 36} = 7$ ; direction cosines $\frac{2}{7}, \frac{3}{7}, \frac{6}{7}$ A1
Correct option: (a) $\frac{2}{7}, \frac{3}{7}, \frac{6}{7}$

Q2·1 mark·Multiple choiceAngle between two lines

The angle between two lines with direction ratios $1, 1, 0$ and $1, 0, 1$ is

Q3·1 mark·Assertion–reasonAngle between two lines

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 2}{4} = \frac{y + 1}{6} = \frac{z}{8}$ are parallel.

Reason (R): Two lines are parallel when their direction ratios are proportional.

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerEquation of a line

Find the vector and Cartesian equations of the line through $(2, - 1, 3)$ parallel to $\hat{i} + 2 \hat{j} - 2 \hat{k}$ .

Show worked answer

Answer:

\vec{r} = (2 \hat{i} - \hat{j} + 3 \hat{k}) + λ (\hat{i} + 2 \hat{j} - 2 \hat{k})

$\vec{r} = (2 \hat{i} - \hat{j} + 3 \hat{k}) + λ (\hat{i} + 2 \hat{j} - 2 \hat{k})$ A1
$\begin{aligned} \frac{x - 2}{1} & = \frac{y + 1}{2} \\ = \frac{z - 3}{- 2} \end{aligned}$ A1

Q5·2 marks·Short answerAngle between two lines

Find $k$ if the lines $\frac{x - 1}{2} = \frac{y + 2}{k} = \frac{z}{3}$ and $\frac{x + 3}{k} = \frac{y - 1}{- 1} = \frac{z - 2}{1}$ are perpendicular.

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerAngle between two lines

Find the angle between the lines $\vec{r} = (\hat{i} + 2 \hat{j}) + λ (2 \hat{i} + 2 \hat{j} + \hat{k})$ and $\vec{r} = (3 \hat{i} - \hat{k}) + μ (2 \hat{i} + \hat{j} - 2 \hat{k})$ .

Show worked answer

Answer:

\cos^{- 1} \frac{4}{9}

${\vec{b}}_{1} \cdot {\vec{b}}_{2} = 4 + 2 - 2 = 4$ M1
$| {\vec{b}}_{1} | = | {\vec{b}}_{2} | = 3$ , so $\cos θ = \frac{4}{9}$ A1
$θ = \cos^{- 1} \frac{4}{9}$ A1

Q7·3 marks·Short answerEquation of a line

Find the points on the line $\frac{x + 2}{3} = \frac{y + 1}{2} = \frac{z - 3}{2}$ that are $5$ units from the point $(1, 3, 3)$ .

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerShortest distance between lines

The skew lines $\vec{r} = (\hat{i} + 2 \hat{k}) + λ (\hat{i} + \hat{j})$ and $\vec{r} = (\hat{j} - \hat{k}) + μ (\hat{j} + \hat{k})$ never meet. How far apart are they at their closest?

Show worked answer

Answer:

\frac{5 \sqrt{3}}{3}

units

$\begin{aligned} {\vec{b}}_{1} \times {\vec{b}}_{2} & = (\hat{i} + \hat{j}) \times (\hat{j} + \hat{k}) \\ = \hat{i} - \hat{j} + \hat{k} \end{aligned}$ M1
$| {\vec{b}}_{1} \times {\vec{b}}_{2} | = \sqrt{3}$ A1
${\vec{a}}_{2} - {\vec{a}}_{1} = - \hat{i} + \hat{j} - 3 \hat{k}$ A1
$\begin{aligned} d & = \frac{| ({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2}) |}{| {\vec{b}}_{1} \times {\vec{b}}_{2} |} \\ = \frac{| - 1 - 1 - 3 |}{\sqrt{3}} \end{aligned}$ M1
$= \frac{5}{\sqrt{3}} = \frac{5 \sqrt{3}}{3}$ units A1

Q9·5 marks·Long answerEquation of a line

Find the foot of the perpendicular from $P (4, 3, - 2)$ to the line $\frac{x - 1}{2} = \frac{y + 1}{1} = \frac{z - 2}{- 2}$ , the perpendicular distance of $P$ from the line, and the image of $P$ in the line.

Section E style · Case study (4 marks)

Q10·4 marks·Case studyShortest distance between lines

Two drone paths. Two drones fly along straight paths $L_{1} : \vec{r} = λ (\hat{i} + \hat{j})$ and $L_{2} : \vec{r} = 2 \hat{k} + μ (\hat{i} - \hat{j})$ (units: metres).

Write the direction ratios of $L_{1}$ . [1 mark]
Are the paths parallel? Give a reason. [1 mark]
How close do the two flight paths come to each other? [2 marks]

Show worked answer

$1, 1, 0$ A1
No: $1, 1, 0$ and $1, - 1, 0$ are not proportional A1
${\vec{b}}_{1} \times {\vec{b}}_{2} = - 2 \hat{k}$ , ${\vec{a}}_{2} - {\vec{a}}_{1} = 2 \hat{k}$ M1
$\begin{aligned} d & = \frac{| 2 \hat{k} \cdot (- 2 \hat{k}) |}{2} \\ = 2 \end{aligned}$ m A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

Find direction cosines/ratios and write vector and Cartesian equations of a line through a point or two points.

Read first: 1. Direction cosines and ratios; 2. Equation of a line 5 practice questions · checkpoint: 3 questions, 5 marks, pass 80%

Practise step 1

Step 2

Board standard

Convert between forms, find angles between lines, test parallel/perpendicular/coincident lines and distances between parallel lines.

Read first: 2. Equation of a line; 3. Angle between two lines; 4. Shortest distance (parallel lines) 24 practice questions · checkpoint: 4 questions, 8 marks, pass 75%

Practise step 2

Step 3

Full marks on long answers

Score full marks on 5-mark shortest-distance, intersection and foot-of-perpendicular/image questions and on 3-D case studies.

Read first: 4. Shortest distance; Worked examples 2-3; Board-exam tips 5 practice questions · checkpoint: 3 questions, 14 marks, pass 70%

Practise step 3

Step 4

95+ stretch (HOTS)

Solve common-perpendicular, image-of-a-point and multi-part triangle problems in 3-D.

Read first: 4. Shortest distance (foot of perpendicular, image) 6 practice questions · checkpoint: 3 questions, 13 marks, pass 60%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 3 of the 46 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceDirection cosines and direction ratios of a line

The direction cosines of the line through $A (1, - 2, 3)$ and $B (3, 1, - 3)$ , directed from $A$ to $B$ , are

(a) $\frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}}, - \frac{6}{\sqrt{13}}$
(b) $2, 3, - 6$
(c) $- \frac{2}{7}, - \frac{3}{7}, \frac{6}{7}$
(d) $\frac{2}{7}, \frac{3}{7}, - \frac{6}{7}$

Q2·1 mark·Multiple choiceEquation of a line (vector and Cartesian)

The Cartesian equations of the line through $(2, - 1, 4)$ parallel to the $y$ -axis are

(a) $\frac{x}{2} = \frac{y}{- 1} = \frac{z}{4}$
(b) $\frac{x - 2}{1} = \frac{y + 1}{0} = \frac{z - 4}{1}$
(c) $\frac{x + 2}{0} = \frac{y - 1}{1} = \frac{z + 4}{0}$
(d) $\frac{x - 2}{0} = \frac{y + 1}{1} = \frac{z - 4}{0}$

Q3·1 mark·Assertion–reasonAngle between two lines

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The lines with direction ratios $1, - 2, 2$ and $2, 2, 1$ are perpendicular.

Reason (R): Two lines are perpendicular if and only if their direction ratios are proportional.

(a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c)Assertion (A) is true but Reason (R) is false.
(d)Assertion (A) is false but Reason (R) is true.

Worked solution·2 marks·Very short answerDirection cosines and direction ratios of a line

Write the line $\frac{3 - x}{2} = \frac{y + 1}{4} = \frac{2 z - 1}{6}$ in standard form and find its direction cosines.

Full worked solution Free, no sign-in

$\frac{x - 3}{- 2} = \frac{y + 1}{4} = \frac{z - \frac{1}{2}}{3}$ : point $(3, - 1, \frac{1}{2})$ , direction ratios $- 2, 4, 3$ M1
Direction cosines $\pm (- \frac{2}{\sqrt{29}}, \frac{4}{\sqrt{29}}, \frac{3}{\sqrt{29}})$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original geometry problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S15 · Problem S71 · Problem S78 · Problem S21. All 22 →

Knowledge organiser: Three Dimensional Geometry on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Three Dimensional Geometry

Taking direction ratios from a non-standard form such as (3 − x)/2 or (2z − 1)/6. Fix: rewrite as (x − 3)/(−2) and (z − ½)/3 first.
Using the skew-line formula for parallel lines. Fix: check proportional direction ratios first; for parallel lines use |b × (a₂ − a₁)|/|b|.
Sign or arithmetic error in b₁ × b₂ that ruins the 5-mark SD question. Fix: check that b₁ × b₂ is perpendicular to both b₁ and b₂ (two dot products).
Concluding that two lines intersect from two coordinates only. Fix: substitute λ, μ into the third coordinate and state the point.
Forgetting the modulus, leaving a negative distance or an obtuse angle between lines. Fix: write |…| in the formula line.
Stopping at the foot of the perpendicular when the image is asked. Fix: image = 2(foot) − point; distance = |point − foot|.

Exam strategy: how answers are step-marked · competency-based questions

Examiner Insights: common mistakes in CBSE Class 12 Maths, with fixes (our analysis of public sources) →

Three Dimensional Geometry in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 14, 19, 30, 35
Sample paper 2 (Board standard): questions 17, 18, 25, 30
Sample paper 3 (Above board): questions 15, 19, 30, 35
Sample paper 4 (Above board): questions 17, 18, 30, 34
Sample paper 5 (95+ challenge): questions 14, 15, 30, 35

Want it against the clock? Take a timed 30-mark chapter test on Three Dimensional Geometry, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on vectors and three-dimensional geometry.

Three Dimensional Geometry: common questions

What is in CBSE Class 12 Maths Chapter 11, Three Dimensional Geometry?

The topics are: Direction cosines and direction ratios of a line, Equation of a line (vector and Cartesian), Angle between two lines, Skew lines and shortest distance, Perpendicular from a point to a line. This page has revision notes for each, and 46 board-style practice questions with step-marked solutions.

How many marks is Three Dimensional Geometry worth?

It is in the Vectors and Three-Dimensional Geometry unit, which carries 14 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Three Dimensional Geometry?

Taking direction ratios from a non-standard form such as (3 − x)/2 or (2z − 1)/6. Fix: rewrite as (x − 3)/(−2) and (z − ½)/3 first. Using the skew-line formula for parallel lines. Fix: check proportional direction ratios first; for parallel lines use |b × (a₂ − a₁)|/|b|.

Are the Three Dimensional Geometry important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Three Dimensional Geometry — revision notes

1. Direction cosines and ratios

2. Equation of a line

3. Angle between two lines

4. Shortest distance

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Three Dimensional Geometry Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Three Dimensional Geometry

Three Dimensional Geometry in our sample papers

Three Dimensional Geometry: common questions

Work through the Route to 95 with your progress saved