3D Geometry — NDA Mathematics Formulas and Traps

Foundations: Coordinates, Distance & Section in Space

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 $B$

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)

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

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)

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

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.

Common traps

Don't forget the square root — or any one of the three squared terms

The distance is

\sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}

, not the bare sum

(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2

(that's

AB^2

). In 3D it is easy to carry only two of the three coordinate differences — Pythagoras runs over ALL THREE axes here, so the

z

-term must be included. Leaving out the

\sqrt{}

gives the squared distance; dropping a term gives a too-small answer.

The ratio m : n weights the FAR endpoint by m — don't swap the weights

For

P

dividing

AB

in ratio

m:n

, the coordinate is

\dfrac{m x_2 + n x_1}{m+n}

— the larger weight

m

multiplies

x_2

(the point

P

is NEARER to

B

). Writing

\dfrac{m x_1 + n x_2}{m+n}

silently divides in ratio

n:m

and lands you on the wrong point. Also:

m:n

is the midpoint ONLY when

m=n

— never average the endpoints for a

2:1

or

1:3

split.

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.

Direction Cosines & Direction Ratios

Learn this subtopic in the notes

Direction ratios, direction cosines, and the unit identity

Direction cosines square-sum to 1

l^2 + m^2 + n^2 = 1, \quad l = \cos\alpha,\ m = \cos\beta,\ n = \cos\gamma

$l,m,n$ direction cosines
$\alpha,\beta,\gamma$ angles with the x, y, z axes

Angle between two lines

Angle between two lines (direction ratios)

\cos\theta = \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 and parallel conditions

Perpendicularity condition

a_1 a_2 + b_1 b_2 + c_1 c_2 = 0

Projection of a segment on an axis or line

Projection of AB on a line of direction cosines ⟨l, m, n⟩

\text{proj} = (x_2-x_1)\,l + (y_2-y_1)\,m + (z_2-z_1)\,n

Direction-angle identities

Core identities

\sum \cos^2\!\theta = 1, \quad \sum \sin^2\!\theta = 2, \quad \sum \cos 2\theta = -1

Direction cosines of the axes and special lines

Line	Direction cosines	Note
x-axis	$\langle 1, 0, 0 \rangle$	makes 0° with x, 90° with y and z
y-axis	$\langle 0, 1, 0 \rangle$	DCs $\langle 0,1,0\rangle$ ; DRs e.g. $\langle 0,4,0\rangle$
z-axis	$\langle 0, 0, 1 \rangle$	perpendicular to the whole XY-plane
⊥ to z-axis	$n = 0$ , e.g. $\langle 5, 6, 0\rangle$	lies in / parallel to the XY-plane A line perpendicular to the z-axis just needs its z-component zero — the x, y parts are free.

Parallel to an axis → that axis's DCs. Perpendicular to an axis → a zero in that slot.

Common traps

Direction RATIOS are not unique; direction COSINES (almost) are

\langle 2,-1,2\rangle

and

\langle 4,-2,4\rangle

are the same direction. Only after normalising do you get direction cosines — and even then a line has TWO sets (

\pm

) for its two orientations. Sum of squares of ratios is NOT 1; only the cosines satisfy that.

You must NORMALISE direction ratios into cosines — and the identity is = 1, not = 0

Direction ratios

\langle a,b,c\rangle

are NOT direction cosines until you divide each by the magnitude

\sqrt{a^2+b^2+c^2}

. Treating raw ratios like

\langle 2,-1,2\rangle

as cosines is wrong — their square-sum is

9

, not

1

. And the defining identity is

l^2 + m^2 + n^2 = 1

(a UNIT vector), never

= 0

;

= 0

would force all three to vanish.

Mind the coefficient and the sign before reading denominators

2(y+3)

is NOT denominator 2 — it's

\frac{y+3}{1/2}

, so the ratio component is

\tfrac12

. And

1 - z

flips the sign:

\frac{z-1}{-1}

. Read symmetric form only after each variable has coefficient

+1

.

For two LINES use their DIRECTIONS, not normals; divide by BOTH magnitudes

The angle between two lines comes from the dot product of their DIRECTION ratios

\cos\theta = \dfrac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}

. Two common slips: (1) forgetting to divide by the product of the magnitudes (the bare dot product is only

\cos\theta

when both are unit vectors / true direction cosines), and (2) borrowing a plane's normal — that's for plane angles. Lines: directions. Planes: normals.

A line cannot make equal acute angles with all three axes unless it's the diagonal

If

\alpha=\beta=\gamma

, then

3\cos^2\alpha = 1

, so

\cos\alpha = \tfrac{1}{\sqrt3}

(

\approx 54.7°

), NOT

45°

or

60°

. Options offering 45°/60° for the equal-angle case are distractors.

The Straight Line in Space

Learn this subtopic in the notes

Equation of a line — symmetric and two-point forms

Symmetric form of a line

\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} = t

Line parallel to, or lying in, a plane

Parallel/contained condition (direction ⟂ normal)

A a + B b + C c = 0

Common traps

Parallel vs lying-in needs the second check

Direction ⟂ normal (dot = 0) only means the line doesn't tilt toward the plane — it could be strictly parallel (misses) OR contained (lies in). You MUST then test a point: on the plane → contained; off → parallel.

direction · normal = 0 means the line is PARALLEL to the plane, not perpendicular

A line is PERPENDICULAR to a plane when its direction is PARALLEL to the normal — the direction ratios are proportional to

\langle A,B,C\rangle

. When the direction-normal dot product is

0

, the line is perpendicular to the NORMAL, hence PARALLEL to (or lying in) the plane. Don't read "dot = 0" as "line ⟂ plane" — it's the exact opposite.

The Plane

Learn this subtopic in the notes

Equation of a plane and its normal

Point-normal form

a(x - x_0) + b(y - y_0) + c(z - z_0) = 0

$\langle a,b,c\rangle$ normal direction ratios
$(x_0,y_0,z_0)$ a point on the plane

Intercept form and special planes

Intercept form

\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1

Plane through three points

Determinant form through three points

\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0

Distance from a point and the foot of the perpendicular

Distance from a point to a plane

\text{distance} = \frac{|a x_1 + b y_1 + c z_1 + d|}{\sqrt{a^2 + b^2 + c^2}}

Angle between two planes

Angle between planes (via normals)

\cos\theta = \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}}

Plane through the line of intersection of two planes

Pencil of planes

P_1 + \lambda P_2 = 0

Common traps

Scale parallel planes to a common normal BEFORE subtracting constants

4x-2y+4z+9=0

and

8x-4y+8z+21=0

look like they differ by 12 in the constant — but the normals differ by a factor of 2. Halve the second plane first; only then is

|d_1-d_2|/|n|

valid.

Always divide by $\sqrt{a^2+b^2+c^2}$ — the numerator alone is NOT the distance

The distance is

\dfrac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}

. Plugging the point into

|ax_1+by_1+cz_1+d|

and stopping there forgets the normalisation by the normal's length — it only equals the distance when

\sqrt{a^2+b^2+c^2}=1

. For

(1,2,2)

and

2x+y+2z+5=0

: numerator

=13

, but the distance is

13/3

, not

13

.

The Sphere

Learn this subtopic in the notes

General equation, centre, and radius

Centre and radius of a sphere

\text{centre} = (-u, -v, -w), \quad r = \sqrt{u^2 + v^2 + w^2 - d}

Diameter form of a sphere

Diameter form

(x-x_1)(x-x_2) + (y-y_1)(y-y_2) + (z-z_1)(z-z_2) = 0

Sphere and a plane — tangency and sections

Tangency condition

p = r \quad\text{(perpendicular distance from centre = radius)}

Sphere and the coordinate axes

Distance from a point to the z-axis

\text{dist to } z\text{-axis} = \sqrt{x_c^2 + y_c^2}

Common traps

The centre is $(-u, -v, -w)$ — NEGATE the half-coefficients

From

x^2+y^2+z^2+2ux+2vy+2wz+d=0

, the centre is

(-u,-v,-w)

, NOT

(u,v,w)

. So a

+2x

term (

2u=2,\ u=1

) puts the centre at

x=-1

, and a

-4x

term (

u=-2

) puts it at

x=+2

. The sign FLIPS. First read

u,v,w

as HALF the linear coefficients, then negate to get the centre.

Radius is $\sqrt{u^2+v^2+w^2-d}$ — mind the $-d$ sign and don't drop the $\sqrt{}$

The radius is

\sqrt{u^2+v^2+w^2-d}

. The constant is SUBTRACTED, so a negative

d

(e.g. the

-2

in

\ldots-2=0

) ADDS to the radical:

-(-2)=+2

. Treating the radius as

u^2+v^2+w^2-d

(forgetting the

\sqrt{}

) gives

r^2

, not

r

; and ensure

x^2,y^2,z^2

each have coefficient 1 before reading

u,v,w,d

.

Touching a PLANE vs touching an AXIS

Tangent to a plane → use the point-to-plane distance for

p

. Tangent to the z-axis → use the distance from the centre to the z-axis,

\sqrt{x_c^2 + y_c^2}

. Don't mix the two distance formulas.

3D Geometry formulas

Foundations: Coordinates, Distance & Section in Space

Section formula — dividing a segment in a ratio

Midpoint and centroid

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

Common traps

Direction Cosines & Direction Ratios

Direction ratios, direction cosines, and the unit identity

Angle between two lines

Perpendicular and parallel conditions

Projection of a segment on an axis or line

Direction-angle identities

Direction cosines of the axes and special lines

Common traps

The Straight Line in Space

Equation of a line — symmetric and two-point forms

Line parallel to, or lying in, a plane

Common traps

The Plane

Equation of a plane and its normal

Intercept form and special planes

Plane through three points

Distance from a point and the foot of the perpendicular

Angle between two planes

Plane through the line of intersection of two planes

Common traps

The Sphere

General equation, centre, and radius

Diameter form of a sphere

Sphere and a plane — tangency and sections

Sphere and the coordinate axes

Common traps

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