NDA Maths · 3D Geometry

Foundations: Coordinates, Distance & Section in Space

Locate points with three coordinates, then measure between them — distance, the dividing point of a segment, midpoints, centroids, and whether points line up.

All 3D Geometry notes NDA Maths strategy

Why this matters

Twenty PYQs across 2017–2026, and the launch pad for everything else in the chapter. Questions test the octant/coordinate-plane setup, the distance formula, the section formula (especially the ratio in which a coordinate plane cuts a segment), centroids, and collinearity / shape tests. Six EASY marks live here every other paper — internalise these five concepts and you bank them on sight.

Concept 1 of 5

The 3D coordinate frame — axes, planes, and octants

Intuition

Add a third axis (the z-axis) straight up out of the familiar xy-plane. Now every point needs three numbers

(x, y, z)

. The three axes are mutually perpendicular and meet at the origin; the three coordinate planes (XY, YZ, ZX) slice all of space into eight corner regions called octants — the 3D version of the four quadrants.

Definition

A point in space is an ordered triple $(x, y, z)$ . The three coordinate planes are:

XY-plane: all points with $z = 0$ .
YZ-plane: all points with $x = 0$ .
ZX-plane: all points with $y = 0$ .

These three planes divide space into 8 octants. The first octant holds points with all three coordinates positive. A point on an axis has its other two coordinates zero; a point on a coordinate plane has exactly one coordinate zero.

Diagram · coordinate planes & octants (drag to rotate)

Three planes (XY, YZ, ZX), each splitting space in two → 2 × 2 × 2 = 8 octants. P sits in the first octant (all coordinates positive).

Location	Condition	Example point
On the x-axis	$y = 0,\ z = 0$	$(5, 0, 0)$
On the XY-plane	$z = 0$	$(3, -2, 0)$
On the YZ-plane	$x = 0$	$(0, 4, 1)$
In the first octant	$x, y, z > 0$	$(2, 3, 4)$ The three coordinate planes (not the three axes) are what divide space — that gives 8 octants, not 6.

Zero coordinates tell you where a point sits: one zero → on a plane, two zeros → on an axis.

Practice this conceptself-check · 4 quick reps

Try it yourself

How many octants do the three coordinate planes divide space into, and how many regions do the two axes of a single plane divide that plane into?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Into how many octants do the coordinate planes divide space?
2.
A point $(0, 5, -2)$ lies on which coordinate plane?
3.
On which axis does $(0, 0, 7)$ lie?
4.
How many coordinates of a point on the x-axis are zero?

From the bank · past-year question

Example 13D GeometryEASY

Into how many compartments do the coordinate planes divide the space?

[Q64 · Apr · 2020]

Concept 2 of 5

Distance between two points

Intuition

The straight-line distance between two points in space is just Pythagoras done twice: take the differences in

x

y

, and

z

, square them, add, and square-root. To find a point's distance from a coordinate axis, drop the coordinate measured along that axis and take the distance of what's left.

Definition

For $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ , the distance $AB$ is the square root of the summed squared coordinate differences. Distance from a point to the x-axis is $\sqrt{y^2 + z^2}$ (ignore $x$ ); similarly $\sqrt{x^2+z^2}$ from the y-axis and $\sqrt{x^2+y^2}$ from the z-axis.

Distance between two points

AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

$(x_1,y_1,z_1)$ coordinates of $A$
$(x_2,y_2,z_2)$ coordinates of $B$

Diagram · magnitude = √(x² + y²)

The components x and y are the legs of a right triangle; the vector is the hypotenuse, so |v| = √(x² + y²) = √(16 + 9) = 5. In 3-D the same idea adds a third leg: |v| = √(x² + y² + z²).

Worked example

Find the distance between

A(1, -2, 3)

and

B(4, 2, 15)

Differences: $\Delta x = 4-1 = 3,\ \Delta y = 2-(-2) = 4,\ \Delta z = 15-3 = 12$ .
Square and add: $3^2 + 4^2 + 12^2 = 9 + 16 + 144 = 169$ .
Square root: $\sqrt{169} = 13$ .

Answer:

AB = 13

Practice this conceptself-check · 4 quick reps

Try it yourself

What is the perpendicular distance from the point

(2, 3, 4)

to the x-axis?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Distance between $(0,0,0)$ and $(2,3,6)$ ?
2.
Distance between $(1,0,0)$ and $(1,4,3)$ ?
3.
Distance from $(3,4,12)$ to the origin?
4.
Distance from $(0,6,8)$ to the y-axis?

From the bank · past-year question

Example 23D GeometryEASY

What is the perpendicular distance from the point

(2,3,4)

to the line

\frac{x-0}{1}=\frac{y-0}{0}=\frac{z-0}{0}

[Q62 · Apr · 2020]

Drill 5 more on distance between two points

Concept 3 of 5

Section formula — dividing a segment in a ratio

Intuition

A point that splits a segment

AB

in the ratio

m:n

is a weighted average of the endpoints — the nearer endpoint gets the larger weight. The classic NDA twist asks in what ratio a coordinate plane cuts a segment: set the relevant coordinate of the dividing point to zero and solve for the ratio.

Definition

The point dividing $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ internally in ratio $m:n$ has each coordinate as the weighted mean below. For external division, replace $n$ with $-n$ . To find where the XY-plane ( $z=0$ ) cuts $AB$ , set the $z$ -coordinate of the dividing point to 0: the ratio is $-z_1 : z_2$ (equivalently $z_1 : z_2$ externally / internally depending on signs).

Internal division in ratio m : n

\left( \frac{m x_2 + n x_1}{m+n},\ \frac{m y_2 + n y_1}{m+n},\ \frac{m z_2 + n z_1}{m+n} \right)

$m:n$ ratio in which the point divides $AB$
$A, B$ the two endpoints

Diagram · section formula (internal vs external), m : n = 2 : 1

Internal: P = (m·b + n·a)/(m + n) sits between A and B. External: Q = (m·b − n·a)/(m − n) sits beyond B — the minus sign is what pushes it outside. The midpoint is the m = n case, (a + b)/2.

Worked example

In what ratio does the XY-plane divide the segment joining

(1, 2, 4)

and

(3, -1, 2)

The XY-plane is $z = 0$ . Let it divide $AB$ in ratio $k:1$ .
The $z$ -coordinate of the dividing point is $\dfrac{k(2) + 1(4)}{k+1}$ .
Set it to 0: $2k + 4 = 0 \Rightarrow k = -2$ .
A negative ratio means external division: the plane divides $AB$ externally in $2:1$ .

Answer:Externally in the ratio

2 : 1

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the coordinates of the point dividing

A(1, -2, 3)

and

B(3, 4, -5)

internally in the ratio

1 : 1

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Point dividing $(0,0,0)$ and $(6,9,3)$ in ratio $2:1$ ?
2.
The XY-plane cuts a segment where which coordinate is set to?
3.
Midpoint of $(2,4,6)$ and $(4,8,10)$ ?
4.
In ratio $k:1$ , the YZ-plane gives which equation?

From the bank · past-year question

Example 33D GeometryMODERATE

The

xy

-plane divides the line segment joining the points

(-1,3,4)

and

(2,-5,6)

[Q63 · Sep · 2021]

Drill 1 more on section formula — dividing a segment in a ratio

Concept 4 of 5

Midpoint and centroid

Intuition

The midpoint averages two points; the centroid of a triangle averages its three vertices. Both are just coordinate-wise means — no ratio bookkeeping. The centroid is the triangle's balance point and divides each median

2:1

Definition

The midpoint of $A$ and $B$ averages the two coordinate triples. The centroid $G$ of triangle $ABC$ averages all three vertices: each coordinate of $G$ is the mean of that coordinate over $A, B, C$ .

Centroid of triangle ABC

G = \left( \frac{x_1+x_2+x_3}{3},\ \frac{y_1+y_2+y_3}{3},\ \frac{z_1+z_2+z_3}{3} \right)

Diagram · closed loop & centroid

Walking the edges A→B→C→A returns you to the start, so AB + BC + CA = 0. The three medians meet at the centroid G = (a + b + c)/3, the average of the vertices' position vectors.

Worked example

Find the centroid of the triangle with vertices

A(1,2,3)

B(4,-1,0)

and

C(7,2,3)

Average the x-coordinates: $(1+4+7)/3 = 12/3 = 4$ .
Average the y-coordinates: $(2-1+2)/3 = 3/3 = 1$ .
Average the z-coordinates: $(3+0+3)/3 = 6/3 = 2$ .

Answer:

G = (4, 1, 2)

Practice this conceptself-check · 4 quick reps

Try it yourself

The midpoint of

A

and

B(4, 2, 8)

(3, 1, 5)

. Find

A

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Centroid of $(0,0,0),(3,0,0),(0,3,3)$ ?
2.
Midpoint of $(1,2,3)$ and $(5,6,7)$ ?
3.
The centroid divides each median in what ratio (vertex : base)?
4.
Centroid of $(2,2,2),(4,4,4),(6,6,6)$ ?

From the bank · past-year question

Example 43D GeometryEASY

The centroid of the triangle with vertices

A(2,-3,3)

B(5,-3,-4)

and

C(2,-3,-2)

is the point

[Q61 · Apr · 2019]

Concept 5 of 5

Collinearity and shape tests

Intuition

Three points are collinear when one segment is a scalar multiple of another — same direction ratios — or, equivalently, when the longest distance equals the sum of the other two. The same distance toolkit classifies triangles (right-angled, isosceles) and quadrilaterals (rectangle, parallelogram): compute side lengths and diagonals and compare.

Definition

Collinearity: $A, B, C$ are collinear iff $\overrightarrow{AB}$ and $\overrightarrow{AC}$ have proportional components, i.e. the same direction ratios. Distance check: collinear iff $AB + BC = AC$ (for $B$ between). Shapes: a triangle is right-angled where two sides satisfy Pythagoras; a parallelogram has equal, bisecting diagonals (midpoint of one diagonal = midpoint of the other); a rectangle additionally has equal diagonals.

Worked example

If the points

A(1, -1, 2)

B(3, k, 4)

and

C(5, 3, 6)

are collinear, find

k

Direction ratios of $\overrightarrow{AC} = C - A = \langle 4, 4, 4\rangle$ , i.e. $\langle 1,1,1\rangle$ .
Direction ratios of $\overrightarrow{AB} = B - A = \langle 2,\ k+1,\ 2\rangle$ .
For collinearity these are proportional: $\dfrac{2}{1} = \dfrac{k+1}{1} = \dfrac{2}{1} = 2$ .
So $k + 1 = 2 \Rightarrow k = 1$ .

Answer:

k = 1

Practice this conceptself-check · 4 quick reps

Try it yourself

Show whether

A(1,2,3)

B(2,3,4)

C(4,5,6)

are collinear using direction ratios.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Are $\langle 2,4,6\rangle$ and $\langle 1,2,3\rangle$ proportional?
2.
Collinearity by distance: which equation holds for B between A and C?
3.
A parallelogram's diagonals do what?
4.
A right angle at B means which sides satisfy Pythagoras?

From the bank · past-year question

Example 53D GeometryMODERATE

If the points

(x, y, -3)

(2, 0, -1)

and

(4, 2, 3)

lie on a straight line, then what are the values of

x

and

y

respectively ?

[Q60 · Sep · 2019]

Collinear vs coplanar vs concyclic

NDA likes asking whether four points are collinear, coplanar, or form a specific shape. Three points are ALWAYS coplanar; the real test is collinearity (proportional direction ratios). For four points, check coplanarity via the scalar triple product of three edge vectors = 0.

Drill 9 more on collinearity and shape tests

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Distance between two points
Distance between two points
$AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$
Section formula — dividing a segment in a ratio
Internal division in ratio m : n
$\left( \frac{m x_2 + n x_1}{m+n},\ \frac{m y_2 + n y_1}{m+n},\ \frac{m z_2 + n z_1}{m+n} \right)$
Midpoint and centroid
Centroid of triangle ABC
$G = \left( \frac{x_1+x_2+x_3}{3},\ \frac{y_1+y_2+y_3}{3},\ \frac{z_1+z_2+z_3}{3} \right)$

Reference tables (1)

The 3D coordinate frame — axes, planes, and octants4 rows

Location	Condition	Example point
On the x-axis	$y = 0,\ z = 0$	$(5, 0, 0)$
On the XY-plane	$z = 0$	$(3, -2, 0)$
On the YZ-plane	$x = 0$	$(0, 4, 1)$
In the first octant	$x, y, z > 0$	$(2, 3, 4)$ The three coordinate planes (not the three axes) are what divide space — that gives 8 octants, not 6.

Zero coordinates tell you where a point sits: one zero → on a plane, two zeros → on an axis.

Watch out for (1)

Collinear vs coplanar vs concyclic
→ Collinearity and shape tests

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 13D GeometryEASY

(0, 0, 0)

(a, 0, 0)

(0, b, 0)

and

(0, 0, c)

are four distinct points. What are the coordinates of the point which is equidistant from the four points?

[Q62 · Apr · 2017]

Example 23D GeometryMODERATE

A point

P

lies on the line joining

A(1,2,3)

and

B(2,10,1)

. If

z

-coordinate of

P

is 7, what is the sum of the other two coordinates?

[Q100 · Sep · 2023]

Example 33D GeometryMODERATE

Consider the following for the items that follow: The position vectors of two points

A

and

B

are

\hat{i}-\hat{j}

and

\hat{j}+\hat{k}

respectively.

Consider the following points: 1.

(-1,-3,1)

(-1,3,2)

(-2,5,3)

Which of the above points lie on the line joining

A

and

B

[Q69 · Apr · 2023]

Example 43D GeometryHARD

Let A=(1,8,4), B=(0,-11,4), C=(2,-3,1). What are the coordinates of D, foot of perpendicular from A to BC?

[Q58 · Apr · 2018]

Example 53D GeometryMODERATE

The points

P(3, 2, 4)

Q(4, 5, 2)

R(5, 8, 0)

and

S(2, -1, 6)

are

[Q63 · Apr · 2017]

Drill every past-year question on this subtopic

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