Introduction to Three Dimensional Geometry Class 11 Notes

Q: What is in CBSE Class 11 Maths Chapter 11, Introduction to Three Dimensional Geometry?

The topics are: Axes, planes and octants, Coordinates of a point, Distance between two points, Using the distance formula. This page has revision notes for each, and 32 board-style practice questions with step-marked solutions.

Q: How many marks is Introduction to Three Dimensional Geometry worth?

It is in the Coordinate Geometry unit, which carries 12 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Q: Where do students lose marks in Introduction to Three Dimensional Geometry?

z-coordinate left out of the distance. Fix: three squared differences, always. Distance from an axis confused with a coordinate. Fix: from the x-axis it is √(y² + z²).

Q: 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.

Revision notes

Introduction to Three Dimensional Geometry: revision notes

1. Axes, planes and octants

Three mutually perpendicular lines through the origin $O$ are the $x$ -, $y$ - and $z$ -axes. They determine three coordinate planes: the $x y$ -plane ( $z = 0$ ), the $y z$ -plane ( $x = 0$ ) and the $z x$ -plane ( $y = 0$ ). The planes divide space into eight octants, numbered I to VIII: I $(+, +, +)$ , II $(-, +, +)$ , III $(-, -, +)$ , IV $(+, -, +)$ , and V to VIII the same with $z < 0$ .

2. Coordinates of a point

$P (x, y, z)$ : $x$ , $y$ , $z$ are the signed distances of $P$ from the $y z$ -, $z x$ - and $x y$ -planes.

On the $x$ -axis: $(x, 0, 0)$ ; on the $y$ -axis: $(0, y, 0)$ ; on the $z$ -axis: $(0, 0, z)$ .
In the $x y$ -plane: $(x, y, 0)$ , and so on. The foot of the perpendicular from $(a, b, c)$ to the $x y$ -plane is $(a, b, 0)$ .

3. Distance between two points

$P Q = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}$

From the origin: $\sqrt{x^{2} + y^{2} + z^{2}}$ .
Distance of $(a, b, c)$ from the $x$ -axis: $\sqrt{b^{2} + c^{2}}$ (from the foot $(a, 0, 0)$ ); from the $x y$ -plane: $| c |$ .

4. Using the distance formula

Collinear points: $A B + B C = A C$ for the right order.
Isosceles / equilateral / right triangles, parallelograms and rectangles: compare sides (and diagonals).
Equidistant points and loci: set squared distances equal; the points at distance $r$ from $(a, b, c)$ satisfy $(x - a)^{2} + (y - b)^{2} + (z - c)^{2} = r^{2}$ .

Worked example 1

Find the distance between $(2, - 1, 3)$ and $(- 2, 1, 3)$ .

$\sqrt{16 + 4 + 0} = \sqrt{20} = 2 \sqrt{5}$ .

Worked example 2

Name the octant of $(4, - 2, - 3)$ .

Signs $(+, -, -)$ : octant VIII.

Worked example 3

Find the point on the $y$ -axis equidistant from $(1, 1, 0)$ and $(0, 3, 1)$ .

$(0, y, 0)$ : $1 + (y - 1)^{2} = (y - 3)^{2} + 1 \Rightarrow 4 y = 8$ , so $(0, 2, 0)$ .

Common errors

Forgetting the $z$ term in the distance formula.
Distance from the $x$ -axis taken as $| a |$ : it is $\sqrt{b^{2} + c^{2}}$ .
Confusing "on the $x$ -axis" $(x, 0, 0)$ with "in the $x y$ -plane" $(x, y, 0)$ .
Claiming collinearity from two equal distances instead of $A B + B C = A C$ .

Exam tips

Write the differences in brackets first, $(4 - 1)^{2} + (2 - (- 2))^{2} + \dots$ , then square.
For "show that", end with the relation you proved (for example $A B^{2} + A C^{2} = B C^{2}$ ).

Topics in this chapter: Axes, planes and octants · Coordinates of a point · Distance between two points · Using the distance formula.

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

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

You can place points in octants and on axes and planes, and use the distance formula.

Read first: 1. Axes, planes and octants; 2. Coordinates; 3. Distance 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%

Practise step 1

Step 2

Exam standard

You can find unknown coordinates from distances and test triangles and collinearity.

Read first: 3. Distance; 4. Using the distance formula; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can classify quadrilaterals in space and handle loci with the distance formula.

Read first: 4. Parallelograms and loci 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can find distances from axes and loci of equidistant points.

Read first: 3. Distance from an axis; 4. Equidistant points; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

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. 2 of the 32 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceAxes, planes and octants

The point $(- 2, 3, - 5)$ lies in octant

(a)II
(b)VI
(c)III
(d)VII

Q2·1 mark·Multiple choiceCoordinates of a point

Every point on the $y$ -axis has coordinates of the form

(a) $(x, 0, 0)$
(b) $(x, y, 0)$
(c) $(0, 0, z)$
(d) $(0, y, 0)$

Q3·1 mark·Multiple choiceDistance between two points

The distance between $(1, - 2, 3)$ and $(4, 2, 3)$ is

(a) $\sqrt{7}$
(b) $7$
(c) $5$
(d) $25$

Worked solution·2 marks·Very short answerUsing the distance formula

Show that $A (1, 2, 3)$ , $B (3, 4, 4)$ and $C (0, 4, 5)$ are the vertices of an isosceles triangle.

Full worked solution Free, no sign-in

$A B = \sqrt{4 + 4 + 1} = 3$ , $A C = \sqrt{1 + 4 + 4} = 3$ M1
$A B = A C$ (and $B C = \sqrt{10}$ ), so the triangle is isosceles 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 S78 · Problem S21 · Problem S77 · Problem S20. All 22 →

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

Where marks are lost in Introduction to Three Dimensional Geometry

z-coordinate left out of the distance. Fix: three squared differences, always.
Distance from an axis confused with a coordinate. Fix: from the x-axis it is √(y² + z²).
Octant named from two signs only. Fix: read all three signs and use the table I to VIII.
Collinearity asserted without the sum of two distances equalling the third. Fix: show AB + BC = AC.
Parallelogram called a rectangle without comparing diagonals. Fix: equal diagonals are needed.

Test Introduction to Three Dimensional Geometry against the clock

Want it against the clock? Take a timed 30-mark chapter test on Introduction to 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.

Introduction to Three Dimensional Geometry: common questions

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