MathsISC Class 11 · Introduction to Three-dimensional Geometry

3D Coordinates, Octants and the Distance Formula: Collinearity and Triangle Problems

Updated 17 October 2026 · Fact-checked

In space a point is given by (x, y, z) with respect to three mutually perpendicular axes; the three coordinate planes divide space into 8 octants. The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²], and the distance of P from the origin is √(x² + y² + z²).

Notice the right angle at D: OD² = x²+y², then OP² = OD² + z², so Pythagoras is used twice to get the distance.

Illustration drawn for QuestionBuddy.

Understand Coordinates, Octants and the Distance Formula in Space

Axes and planes. The x-, y- and z-axes meet at the origin and are mutually perpendicular. The coordinate planes are the xy-plane (z = 0), yz-plane (x = 0) and zx-plane (y = 0). They divide space into eight octants.

Signs in the octants. Octant I: (+, +, +); II: (−, +, +); III: (−, −, +); IV: (+, −, +); V: (+, +, −); VI: (−, +, −); VII: (−, −, −); VIII: (+, −, −).

Special points. On the x-axis: (x, 0, 0); on the y-axis: (0, y, 0); on the z-axis: (0, 0, z). In the xy-plane: (x, y, 0). The distance of P(x, y, z) from the xy-plane is |z|, from the yz-plane |x|, and from the zx-plane |y|. The foot of the perpendicular from P to the x-axis is (x, 0, 0).

Distance formula. $P Q = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$ , from applying Pythagoras twice. Three points are collinear if the largest of the three distances equals the sum of the other two.

Shapes. To prove that three points form an isosceles, equilateral or right-angled triangle, compute the three side lengths and compare their squares. A parallelogram has equal opposite sides and bisecting diagonals.

Locus on an axis. A point equidistant from two given points and on the y-axis has coordinates (0, y, 0); equate the squared distances and solve for y.

Coordinates, Octants and the Distance Formula in Space: formulas to remember

Distance: $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$; From the origin: $x^{2} + y^{2} + z^{2}$ .
Distances from the planes: $xy-plane: $|z|$ | yz-plane: $|x|$ | zx-plane: $|y|$$; Coordinates of the foot: replace that coordinate by 0.
Octant signs: $I (+, +, +) ∣ I I (-, +, +) ∣ I I I (-, -, +) ∣ I V (+, -, +) ∣ V (+, +, -) ∣ V I (-, +, -) ∣ V I I (-, -, -) ∣ V I I I (+, -, -)$; Octant I is the all-positive one.
Collinearity: $L a r g es t d i s t an ce = s u m o f t h eo t h er tw o$; Otherwise a triangle forms.

Coordinates, Octants and the Distance Formula 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.

A drone to a warehouse

Suppose a drone at (1, 2, 3) metres is to fly in a straight line to a landing pad at (4, 6, 15).

Find the distance flown and the drone's height above the ground plane (xy-plane) at the start.

Try it first, then show the solutionSolution

$P Q = (4 - 1)^{2} + (6 - 2)^{2} + (15 - 3)^{2} = 9 + 16 + 144 = 169$ .
$P Q = 13$ m.
Height above the xy-plane at the start is |z| = 3 m.

Answer: 13 m; initial height 3 m.

Memory hook: Add the squares of the three differences.

Tricks to remember Coordinates, Octants and the Distance Formula in Space

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

Signs decide the octant
List the signs of x, y and z in order.
Use it when: Octant questions.
Compare squares
To check a right angle, test the squares of the sides.
Use it when: Triangle problems.

Coordinates, Octants and the Distance Formula in Space: solved examples

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

Example 1: Distance between two points · 1 mark · MCQBoard level

The distance between (1, 2, 3) and (4, 6, 15) is:
(a) 11
(b) 12
(c) 13
(d) 14

Try it first, then show the solutionSolution

$9 + 16 + 144 = 169 = 13$ .

Answer: (c) 13

Example 2: Octant and distances · 3 marks · Short answerBoard level

In which octant does (−2, 3, −4) lie? Find its distances from the origin and from each coordinate plane.

Try it first, then show the solutionSolution

The signs (−, +, −) give octant VI. (1 mark)
Distance from the origin $= 4 + 9 + 16 = 29$ . (1 mark)
From the xy-plane: 4; yz-plane: 2; zx-plane: 3. (1 mark)

Answer: Octant VI; √29; planes: 4, 2, 3.

Example 3: Right-angled triangle · 4 marks · Short answerHarder board level

Show that the points A(0, 7, −10), B(1, 6, −6) and C(4, 9, −6) are the vertices of an isosceles right-angled triangle.

Try it first, then show the solutionSolution

$A B = 1 + 1 + 16 = 18$ . (1 mark)
$B C = 9 + 9 + 0 = 18$ and $A C = 16 + 4 + 16 = 6$ . (1.5 marks)
$A B = B C$ (isosceles). (0.5 mark)
$A B^{2} + B C^{2} = 18 + 18 = 36 = A C^{2}$ , so the angle at B is a right angle. (1 mark)

Answer: AB = BC = 3√2, AC = 6, and AB² + BC² = AC².

Example 4: Collinear points · 4 marks · Short answerHarder board level

Show that P(1, 2, 3), Q(2, 4, 5) and R(4, 8, 9) are collinear. Find the point on the y-axis equidistant from (3, 1, 2) and (5, 5, 2).

Try it first, then show the solutionSolution

$P Q = 1 + 4 + 4 = 3$ , $QR = 4 + 16 + 16 = 6$ , $P R = 9 + 36 + 36 = 9$ . (1.5 marks)
$P Q + QR = 9 = P R$ , so they are collinear. (0.5 mark)
Let the point be (0, y, 0): $9 + (y - 1)^{2} + 4 = 25 + (y - 5)^{2} + 4$ . (1 mark)
$14 - 2 y = 54 - 10 y$ , so $y = 5$ : the point is (0, 5, 0). (1 mark)

Answer: Collinear; (0, 5, 0).

Example 5: Parallelogram in space · 5 marks · Long answerHOTS / competency

Show that A(2, 3, 5), B(5, 1, 6), C(6, 4, 9), D(3, 6, 8) form a parallelogram, and decide whether it is a rhombus.

Try it first, then show the solutionSolution

$A B = 9 + 4 + 1 = 14$ ; $C D = 9 + 4 + 1 = 14$ . (1 mark)
$B C = 1 + 9 + 9 = 19$ ; $D A = 1 + 9 + 9 = 19$ . (1 mark)
Midpoint of AC $= (4, 3.5, 7)$ ; midpoint of BD $= (4, 3.5, 7)$ : the diagonals bisect each other. (1.5 marks)
It is a parallelogram. (0.5 mark)
The adjacent sides $14 \neq = 19$ , so it is not a rhombus. (1 mark)

Answer: A parallelogram, not a rhombus.

Where students lose marks in Coordinates, Octants and the Distance Formula in Space

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

Mistake: Wrong octant
Mixing the order of the signs.
Do this: List the signs of x, y, z in order and match.
Usually costs: 1 to 2 marks.
Mistake: Leaving out the third coordinate
Using the 2D distance formula.
Do this: Add (z₂ − z₁)².
Usually costs: 2 marks.
Mistake: Not squaring before comparing
Comparing the side lengths for a right angle.
Do this: Check a² + b² = c².
Usually costs: 1 mark.

Exam tips

Write the differences before squaring.
Keep the exact surd until the end.
Use midpoint checks for parallelograms.
Draw a rough sketch of the points.

Practice questions on Coordinates, Octants and the Distance Formula 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 point (3, −2, −5) lies in the octant:
(a) IV
(b) V
(c) VIII
(d) VII
Hint
Signs (+, −, −).
Answer
(c) VIII
Q2 · 2 marks · Short answerBoard level
Find the distance of (3, 4, 12) from the origin.
Hint
Pythagoras in space.
Answer
$9 + 16 + 144 = 13$ .
Q3 · 3 marks · Short answerHarder board level
Find the lengths of the sides of the triangle with vertices (1, −1, 0), (3, 1, 2) and (5, −3, 4) and decide the type.
Hint
Compute all three sides.
Answer
$A B = 4 + 4 + 4 = 23$ , $B C = 4 + 16 + 4 = 26$ , $A C = 16 + 4 + 16 = 6$ . $A B^{2} + A C^{2} = 12 + 36 = 48 \neq = 24$ and $A B^{2} + B C^{2} = 12 + 24 = 36 = A C^{2}$ , so the angle at B is a right angle (scalene right triangle).
Q4 · 3 marks · Short answerHarder board level
Find the point on the z-axis equidistant from (1, 5, 7) and (5, 1, −4).
Hint
Let the point be (0, 0, z).
Answer
$1 + 25 + (z - 7)^{2} = 25 + 1 + (z + 4)^{2}$ gives $- 14 z + 49 = 8 z + 16$ , so $z = \frac{33}{22} = 1.5$ : point (0, 0, 1.5).
Q5 · 3 marks · Short answerHarder board level
If the distance of (x, 2, 3) from (1, −1, 4) is $19$ , find x.
Hint
Set up the equation.
Answer
$(x - 1)^{2} + 9 + 1 = 19$ gives $(x - 1)^{2} = 9$ , so $x = 4$ or $x = - 2$ .
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a surveyor models a hill with the base point O(0, 0, 0) and two flags at A(6, 8, 0) and B(6, 8, 24) (metres).
(i) Find OA. (1 mark)
(ii) Find AB. (1 mark)
(iii) Find OB. (1 mark)
(iv) Check OB² = OA² + AB². (1 mark)
Hint
AB is vertical.
Answer
(i) $O A = 36 + 64 = 10$ m. (ii) $A B = 24$ m. (iii) $O B = 36 + 64 + 576 = 26$ m. (iv) $676 = 100 + 576$ ✓.

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.Number of octants?Show answer
8.
2.Distance from the xy-plane?Show answer
|z|.
3.Distance from the origin?Show answer
√(x² + y² + z²).
4.Point on the y-axis?Show answer
(0, y, 0).

Coordinates, Octants and the Distance Formula in Space: frequently asked questions

Why are there 8 octants?

Each of the three coordinates can be positive or negative: 2 × 2 × 2 = 8 combinations.

What is the equation of the xy-plane?

z = 0.

Is the distance formula symmetric?

Yes, PQ = QP, since the differences are squared.