MathsCBSE Class 11 · Introduction to Three Dimensional Geometry

Octants, Projections and Reflections of a Point in Space (Class 11)

Updated 17 October 2026 · Fact-checked

The three coordinate planes divide space into eight octants, fixed by the signs of (x, y, z). The distance of P(x, y, z) from the YZ, ZX and XY planes is |x|, |y| and |z|; its distance from the x-, y- and z-axes is √(y² + z²), √(x² + z²) and √(x² + y²). Reflecting in a coordinate plane changes the sign of the one coordinate that is perpendicular to that plane.

Dropping P straight down gives its projection (x, y, 0); continuing the same distance below the XY-plane gives the image P' where only z changes sign.

Illustration drawn for QuestionBuddy.

Understand Octants, Projections and Reflections of Points in Space

Fix three mutually perpendicular axes $O X$ , $O Y$ , $O Z$ meeting at the origin $O$ . A point $P$ has coordinates $(x, y, z)$ , where $x, y, z$ are its signed distances from the $Y Z$ , $Z X$ and $X Y$ planes respectively.

Octants. The coordinate planes cut space into eight octants:

Octant	I	II	III	IV	V	VI	VII	VIII
$(x, y, z)$	$+ + +$	$- + +$	$- - +$	$+ - +$	$+ + -$	$- + -$	$- - -$	$+ - -$

Octants I to IV lie above the $X Y$ -plane ( $z > 0$ ), V to VIII below it. To place a point, read off the signs; to count octants for a condition such as $x y > 0, z < 0$ , count the allowed sign patterns (here $x, y$ both positive or both negative, $z$ negative: 2 octants).

Points on the axes and planes. A point on the $x$ -axis has the form $(a, 0, 0)$ ; on the $y$ -axis $(0, b, 0)$ ; on the $z$ -axis $(0, 0, c)$ . A point in the $X Y$ -plane has $z = 0$ , in the $Y Z$ -plane $x = 0$ , in the $Z X$ -plane $y = 0$ .

Distances from planes and axes. For $P (x, y, z)$ :

from the $Y Z$ , $Z X$ , $X Y$ planes: $∣ x ∣$ , $∣ y ∣$ , $∣ z ∣$ ;
from the $x$ -axis: $y^{2} + z^{2}$ ; from the $y$ -axis: $x^{2} + z^{2}$ ; from the $z$ -axis: $x^{2} + y^{2}$ (each is the distance to the foot of the perpendicular, e.g. $(x, 0, 0)$ for the $x$ -axis).

Feet of perpendiculars and projections. The foot on the $x$ -axis is $(x, 0, 0)$ ; the projection on the $X Y$ -plane is $(x, y, 0)$ , on the $Y Z$ -plane $(0, y, z)$ , on the $Z X$ -plane $(x, 0, z)$ .

Reflections. The image in the $X Y$ -plane is $(x, y, - z)$ , in the $Y Z$ -plane $(- x, y, z)$ , in the $Z X$ -plane $(x, - y, z)$ . A point and its mirror image are the same distance from the origin.

Cuboid vertices. If one vertex of a box with edges along the axes is the origin and the opposite vertex is $(a, b, c)$ , the other six vertices are obtained by choosing each coordinate as $0$ or its full value: $(a, 0, 0), (0, b, 0), (0, 0, c), (a, b, 0), (a, 0, c), (0, b, c)$ . The space diagonal has length $a^{2} + b^{2} + c^{2}$ , from the distance formula.

Octants, Projections and Reflections of Points in Space: formulas to remember

Distance from the coordinate planes: $∣ x ∣ f r o m t h e Y Z - pl an e; ∣ y ∣ f r o m t h e Z X - pl an e; ∣ z ∣ f r o m t h e X Y - pl an e$; Always non-negative.
Distance from the axes: $\sqrt (y^{2} + z^{2}) f r o m t h e x - a x i s; \sqrt (x^{2} + z^{2}) f r o m t h ey - a x i s; \sqrt (x^{2} + y^{2}) f r o m t h ez - a x i s$; Drop the coordinate along the axis.
Reflection in the XY-plane: $(x, y, z) g oes t o (x, y, - z)$; Similarly for the other planes.
Projection on the XY-plane: $(x, y, 0)$; Foot of the perpendicular from P.
Space diagonal of a cuboid: $\sqrt (a^{2} + b^{2} + c^{2})$; From the origin to (a, b, c).

How to solve Octants, Projections and Reflections of Points in Space questions

Use these steps for octant, distance, projection and reflection questions.

1Write the point as $(x, y, z)$ and note the sign of each coordinate.
2For the octant, match the sign pattern with the table (positive $z$ means octants I to IV).
3For distance from a plane, take the absolute value of the coordinate that is zero on that plane.
4For distance from an axis, drop the coordinate along that axis and take $sum of squares of the other two$ .
5For a reflection, change the sign of only the coordinate perpendicular to the mirror plane.
6For a box, list vertices by choosing 0 or the full value for each coordinate.

Octants, Projections and Reflections of Points in Space in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Octants as above and below ground

Suppose a mine map uses the entrance as origin, $x$ east, $y$ north and $z$ upward. A sensor sits at $(- 2, 5, - 3)$ metres.

In which octant is it, and how far is it from the vertical line through the entrance (the $z$ -axis)?

Try it first, then show the solutionSolution

Signs $(-, +, -)$ give octant VI, below the ground plane.
Distance from the $z$ -axis $= x^{2} + y^{2} = 4 + 25 = 29 \approx 5.39$ m.
Depth below the ground plane $= ∣ z ∣ = 3$ m.

Answer: Octant VI; about 5.39 m from the vertical axis; 3 m deep.

Memory hook: Below the XY-plane means octants V to VIII.

Reflection as a mirror image

Suppose a drone hovers at $(4, - 2, 6)$ m above a still lake whose surface is the $X Y$ -plane, and its reflection is seen in the water.

Find the coordinates of the reflection and the distance between the drone and its reflection.

Try it first, then show the solutionSolution

The mirror is the $X Y$ -plane, so the image is $(4, - 2, - 6)$ .
Distance $= ∣6 - (- 6) ∣ = 12$ m (only $z$ differs).

Answer: Image $(4, - 2, - 6)$ ; 12 m apart.

Memory hook: Mirror in the plane: flip only the coordinate that points out of it.

Tricks to remember Octants, Projections and Reflections of Points in Space

Short memory aids. Say each one aloud once, then use it on a question the same day.

Zero tells the plane
The coordinate that is zero names the plane the point lies in.
Use it when: Points on planes and axes.
Drop the axis letter
Distance from the x-axis ignores x: use y and z.
Use it when: Axis distance questions.
Flip one sign
Reflection changes the sign of just one coordinate.
Use it when: Images in coordinate planes.
Above or below
Positive z: octants I to IV; negative z: V to VIII.
Use it when: Quick octant check.

Octants, Projections and Reflections of Points in Space: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Octant and plane distances · 2 marks · Very short answerBoard level

State the octant of $P (4, - 3, - 2)$ and find its distances from the three coordinate planes.

Try it first, then show the solutionSolution

Signs $(+, -, -)$ give octant VIII. (1 mark)
Distances: from $Y Z$ -plane $∣4∣ = 4$ , from $Z X$ -plane $∣ - 3∣ = 3$ , from $X Y$ -plane $∣ - 2∣ = 2$ . (1 mark)

Answer: Octant VIII; 4, 3 and 2 units

Example 2: Feet, projections and image · 3 marks · Short answerBoard level

For $A (3, - 4, 5)$ write the feet of the perpendiculars on the three axes, the projection on the $X Y$ -plane, the image in the $X Y$ -plane and the distance from the $x$ -axis.

Try it first, then show the solutionSolution

Feet: $(3, 0, 0)$ on the $x$ -axis, $(0, - 4, 0)$ on the $y$ -axis, $(0, 0, 5)$ on the $z$ -axis. (1 mark)
Projection on the $X Y$ -plane: $(3, - 4, 0)$ ; image in the $X Y$ -plane: $(3, - 4, - 5)$ . (1 mark)
Distance from the $x$ -axis $= (- 4)^{2} + 5^{2} = 41$ . (1 mark)

Answer: Feet $(3, 0, 0), (0, - 4, 0), (0, 0, 5)$ ; projection $(3, - 4, 0)$ ; image $(3, - 4, - 5)$ ; $41$

Example 3: A point on an axis · 3 marks · Short answerHarder board level

Find the point on the $y$ -axis equidistant from $A (3, 1, 2)$ and $B (5, 5, 6)$ .

Try it first, then show the solutionSolution

Let the point be $P (0, y, 0)$ . $P A^{2} = 9 + (y - 1)^{2} + 4$ . (½ mark)
$P B^{2} = 25 + (y - 5)^{2} + 36$ . (½ mark)
Equate: $y^{2} - 2 y + 14 = y^{2} - 10 y + 86$ , so $8 y = 72$ , $y = 9$ . (1 mark)
Check: $P A^{2} = 9 + 64 + 4 = 77 = P B^{2} = 25 + 16 + 36$ . Point $(0, 9, 0)$ . (1 mark)

Answer: $(0, 9, 0)$

Example 4: Vertices of a cuboid · 3 marks · Short answerHarder board level

A cuboid has one vertex at the origin, edges along the axes and the opposite vertex at $(3, 4, 12)$ . Write all eight vertices and find the length of the space diagonal and of the diagonal of the base.

Try it first, then show the solutionSolution

Vertices: $(0, 0, 0), (3, 0, 0), (0, 4, 0), (0, 0, 12), (3, 4, 0), (3, 0, 12), (0, 4, 12), (3, 4, 12)$ . (1 mark)
Space diagonal $= 9 + 16 + 144 = 169 = 13$ . (1 mark)
Base (on the $X Y$ -plane) diagonal from $(0, 0, 0)$ to $(3, 4, 0)$ $= 9 + 16 = 5$ . (1 mark)

Answer: Eight vertices as listed; space diagonal 13; base diagonal 5

Example 5: Case study: two drones · 4 marks · Case-basedHOTS / competency

Suppose the launch pad is the origin $O$ with the ground as the $X Y$ -plane (1 unit = 1 m). Drone 1 is at $P (2, 3, 6)$ and drone 2 is at $Q (5, 7, 18)$ .
(i) State the height of drone 1. (1 mark)
(ii) Find its distance from the launch pad. (1 mark)
(iii) Find the point on the ground vertically below drone 1 and the horizontal distance of that point from the pad. (1 mark)
(iv) Find the distance between the two drones. (1 mark)

Try it first, then show the solutionSolution

(i) Height $= ∣ z ∣ = 6$ m. (1 mark)
(ii) $O P = 4 + 9 + 36 = 7$ m. (1 mark)
(iii) Foot $(2, 3, 0)$ ; horizontal distance $4 + 9 = 13 \approx 3.61$ m. (1 mark)
(iv) $P Q = 9 + 16 + 144 = 13$ m. (1 mark)

Answer: (i) 6 m (ii) 7 m (iii) $(2, 3, 0)$ , $13$ m (iv) 13 m

Where students lose marks in Octants, Projections and Reflections of Points in Space

The slips examiners see most often, and what to write instead.

Mistake: Using the wrong coordinate for plane distance
The distance from the XY-plane is |z|, not x or y.
Wrong working
Distance of $(4, - 3, - 2)$ from the $X Y$ -plane taken as 4.
Right working
It is $∣ z ∣ = 2$ .
Do this: The plane named in the question has the coordinate that is NOT in its name.
Usually costs: 1 mark.
Mistake: Keeping the axis coordinate in an axis distance
Students use the full distance formula to the origin.
Wrong working
Distance of $(3, - 4, 5)$ from the $x$ -axis $= 9 + 16 + 25$ .
Right working
$16 + 25 = 41$ .
Do this: The foot of the perpendicular is $(x, 0, 0)$ .
Usually costs: The answer.
Mistake: Changing all signs in a reflection
Mixing a reflection in a plane with a reflection in the origin.
Wrong working
Image of $(3, - 4, 5)$ in the $X Y$ -plane $= (- 3, 4, - 5)$ .
Right working
$(3, - 4, - 5)$ .
Do this: Only the coordinate perpendicular to the mirror flips.
Usually costs: 1 to 2 marks.
Mistake: Wrong octant number
The octant order is not memorised.
Wrong working
$(4, - 3, - 2)$ called octant VII.
Right working
Signs $(+, -, -)$ are octant VIII.
Do this: Write the table of sign patterns in the margin.
Usually costs: 1 mark.

Exam tips

Keep the octant table in a corner of the page: I to IV above the XY-plane, V to VIII below, going round in the same order.
For axis distances write the foot of the perpendicular first; the distance formula then does the rest.
In cuboid questions list all eight vertices systematically.
The section formula in three dimensions is formative only in 2026-27; practise the distance formula instead.
Check a reflection by confirming the midpoint of the point and its image lies in the mirror plane.

Practice questions on Octants, Projections and Reflections of Points in Space

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The image of the point $(2, - 3, 4)$ in the $Y Z$ -plane is:
(a) $(- 2, - 3, 4)$
(b) $(2, 3, 4)$
(c) $(2, - 3, - 4)$
(d) $(- 2, 3, - 4)$
Hint
The mirror is the $Y Z$ -plane, perpendicular to the $x$ -axis.
Answer
(a). Only $x$ changes sign.
Q2 · 1 mark · MCQHarder board level
The number of octants in which a point $(x, y, z)$ with $x y > 0$ and $z < 0$ can lie is:
(a) 1
(b) 2
(c) 4
(d) 8
Hint
List the sign patterns of $x, y$ with $x y > 0$ .
Answer
(b). $x, y$ both positive or both negative, $z$ negative: octants V and VII.
Q3 · 3 marks · Short answerHarder board level
Find the distance of $A (- 2, 3, - 6)$ from each of the three coordinate axes.
Hint
Drop the coordinate along the axis.
Answer
From the $x$ -axis: $9 + 36 = 35$ . From the $y$ -axis: $4 + 36 = 210$ . From the $z$ -axis: $4 + 9 = 13$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The points $(1, 2, 3)$ and $(1, 2, - 3)$ are mirror images in the $X Y$ -plane and are at the same distance from the origin.
Reason (R): Reflection in the $X Y$ -plane changes only the sign of the $z$ -coordinate.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Compare $1 + 4 + 9$ with $1 + 4 + 9$ .
Answer
(a). Both distances are $14$ since $z^{2}$ is the same, and R explains why.
Q5 · 4 marks · Short answerHOTS / competency
A point $P$ lies in octant VII. It is three times as far from the $X Y$ -plane as from the $Y Z$ -plane, and its distance from the $Z X$ -plane equals its distance from the $X Y$ -plane. If $O P = 219$ , find $P$ and its image in the $Z X$ -plane.
Hint
Let the distance from the $Y Z$ -plane be $t$ .
Answer
Octant VII means all coordinates negative. Let $∣ x ∣ = t$ ; then $∣ z ∣ = 3 t$ , $∣ y ∣ = 3 t$ . $O P^{2} = t^{2} + 9 t^{2} + 9 t^{2} = 19 t^{2} = 76$ , so $t = 2$ . $P = (- 2, - 6, - 6)$ . Image in the $Z X$ -plane: $(- 2, 6, - 6)$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.How many octants are there?Show answer
Eight.
2.What is the distance of (x, y, z) from the XY-plane?Show answer
|z|.
3.What is the distance of (x, y, z) from the x-axis?Show answer
√(y² + z²).
4.What is the image of (x, y, z) in the ZX-plane?Show answer
(x, −y, z).
5.In which octant is (−1, 2, −3)?Show answer
VI.

Octants, Projections and Reflections of Points in Space: frequently asked questions

Which octant has all coordinates negative?

Octant VII.

What does a zero coordinate mean?

The point lies in the coordinate plane named by that coordinate being zero. For example z = 0 means the XY-plane.

How is the distance from an axis different from the distance from a plane?

A plane distance is the absolute value of one coordinate. An axis distance is √ of the sum of squares of the other two coordinates.

Is the section formula in 3D examined?

The chapter page notes that the section formula in three dimensions is assessed only formatively in 2026-27, so the annual exam relies on coordinates and the distance formula.