Three Dimensional Geometry Formula Sheet — JEE Main Mathematics

Every key Three Dimensional Geometry formula, definition and theorem for JEE Main Mathematics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 10 sub-topics

Coordinates of a point in space
Distance between two points
Section formula
Direction ratios and direction cosines
Angle between two intersecting lines
Skew lines
Shortest distance between two lines
Equation of a line and a plane in different forms
Intersection of a line and a plane
Coplanarity of two lines

Coordinates, Distance & Direction Cosines/Ratios

Direction cosines & direction ratios: The

direction cosines

l,m,n of a line are the cosines of the angles it makes with the x,y,z axes, and always satisfy

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

Direction ratios

a,b,c are any numbers proportional to l,m,n — a direction vector

a \hat{i} + b \hat{j} + c \hat{k}

Formulas in space

▸Distance: $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$ .
▸Section (m:n): $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n}, \frac{m z _{2} + n z _{1}}{m + n})$ .
▸DRs of the line AB: $(x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})$ .
▸DR $\to$ DC: divide each by $a^{2} + b^{2} + c^{2}$ .

★ Remember · l² + m² + n² = 1, but a² + b² + c² ≠ 1

Direction

cosines

are normalised:

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

. Direction

ratios

are not — they only fix the direction up to a scalar, so

(1, 2, 2)

and

(2, 4, 4)

describe the same line.

🎯 Exam · Two directions from one ratio set

Each direction ratio set gives

two

opposite sets of direction cosines (e.g.

\pm \frac{1}{3} (1, 2, 2)

). For a line either sign is fine; for a directed segment, choose the sign that matches the orientation.

⚠️ Watch out · Sum of direction cosines isn't 1

It is the sum of the

squares

that is

1

(

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

), not

l + m + n

. A common error is to set

l + m + n = 1

— that has no special meaning.

🚫 Examiner Trap · Examiner traps

(1) Direction COSINES always satisfy

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

; direction RATIOS are only PROPORTIONAL (a,b,c) — don't treat DRs as DCs without normalising. (2) Convert via

l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}

etc. (3) A direction has TWO sets of DCs (

\pm

) — opposite senses. (4) DRs of the join of two points are

(x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})

, not their sum.

The Straight Line in Space

Equation of a line: A line is a point on it plus a direction.

Vector

r = a + λ b

(

a

a point,

b

the direction).

Cartesian

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

, where (a,b,c) are direction ratios.

★ Remember · A general point on the line

From the Cartesian form, set each ratio equal to

λ

: every point on the line is

(x_{1} + aλ, y_{1} + bλ, z_{1} + c λ)

. This single parametrisation solves almost every intersection/foot/image problem.

Building the equation

▸Through two points A,B: direction $b = B - A$ , so $r = a + λ (b - a)$ .
▸Through a point with given DRs: substitute straight into the Cartesian form.
▸Convert vector $\to$ Cartesian by reading off the point and the direction components.
▸If a DR is $0$ (say $a = 0$ ): that coordinate is constant, $x = x_{1}$ .

🎯 Exam · Vector ⇔ Cartesian

r = (x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k}) + λ (a \hat{i} + b \hat{j} + c \hat{k})

is exactly

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

. The point gives the constants, the direction gives the denominators.

⚠️ Watch out · Denominators are direction ratios, not points

\frac{x - x _{1}}{a} = \dots

, the numbers a,b,c are the

direction ratios

— not coordinates of a point. Mixing them up (e.g. putting

x_{2}

where a belongs) is a frequent slip.

🚫 Examiner Trap · Examiner traps

(1) In

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

, the denominators are DIRECTION RATIOS and the numerators use a POINT on the line — don't mix them up. (2) A general point on the line is

(x_{1} + aλ, y_{1} + bλ, z_{1} + c λ)

— the key to intersection problems. (3) Through two points A,B: direction

= B - A

. (4) If a DR is

0

(say

a = 0

), that coordinate is FIXED: write

x = x_{1}

separately.

Angle Between Lines, Skew Lines & Shortest Distance

The angle between two lines, a sketch of skew lines, and the shortest-distance formulas for skew and parallel lines with the coplanarity condition — The angle, skew lines, and the shortest distance.

★ Remember · Angle between two lines

cos θ = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}

. Perpendicular

\Leftrightarrow a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0

; parallel

\Leftrightarrow \frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}

🎯 Exam · Shortest distance between skew lines

For

r = a_{1} + λ b_{1}

and

r = a_{2} + μ b_{2}

d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}

. For

parallel

lines use

d = \frac{∣ ( a _{2} - a _{1} ) \times b ∣}{∣ b ∣}

Skew vs intersecting

▸ $Skew$ lines are not parallel and do not meet.
▸Lines are coplanar (intersect or are parallel) $\Leftrightarrow d = 0$ .
▸ $d = 0 \Leftrightarrow (a_{2} - a_{1}) \cdot (b_{1} \times b_{2}) = 0$ (scalar triple product).
▸Cartesian coplanarity: a $3 \times 3$ determinant $= 0$ .

⚠️ Watch out · Use the cross product, not the difference of directions

The shortest distance involves

b_{1} \times b_{2}

(perpendicular to both lines), not

b_{1} - b_{2}

. And the numerator is a

scalar triple product

(a_{2} - a_{1}) \cdot (b_{1} \times b_{2})

— keep the dot and cross in the right places.

🚫 Examiner Trap · Examiner traps

(1) The angle between two lines uses

cos θ

with their direction ratios; perpendicular

\Leftrightarrow a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0

(NOT the cross product). (2) Skew lines are NOT parallel and do NOT intersect — shortest distance

d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}

. (3) For PARALLEL lines use

d = \frac{∣ ( a _{2} - a _{1} ) \times b ∣}{∣ b ∣}

(different formula). (4)

d = 0 \Leftrightarrow

the lines are coplanar (intersect).

The Plane — Equations & Forms

The plane: A plane is fixed by a point on it and a

normal

direction

n = (a, b, c)

. Its general equation is

a x + b y + cz + d = 0

, and the coefficients of x,y,z are exactly the components of the normal vector.

Forms of a plane

Form	Equation
General	$a x + b y + cz + d = 0$ (normal $(a, b, c)$ )
Point-normal	$a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0$
Vector	$r \cdot \overset{n}{^} = p$ ( $\overset{n}{^}$ unit normal, p = distance from O)
Normal form	$l x + m y + n z = p$ ( $l, m, n$ = DCs of normal)
Intercept	$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

The forms

▸General: $a x + b y + cz + d = 0$ ; normal $n = (a, b, c)$ .
▸Point-normal: $a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0$ .
▸Vector / normal: $r \cdot \overset{n}{^} = p$ , i.e. $l x + m y + n z = p$ ( $p =$ distance from origin).
▸Intercept: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ .

★ Remember · The coefficients are the normal

a x + b y + cz + d = 0

, the vector (a,b,c) is

perpendicular

to the plane. This single fact powers angle, distance, parallel and perpendicular conditions — read off the normal first.

🎯 Exam · Plane through three points

Take two edge vectors from one vertex, their cross product is the normal

n

; then use point-normal form. Equivalently, set the

3 \times 3

determinant of the edge vectors (to the general point) to

0

⚠️ Watch out · Normalise before reading p

r \cdot \overset{n}{^} = p

, the normal must be a

unit

vector for p to be the true distance from the origin. If you have a general

a x + b y + cz = d

, divide through by

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

first.

🚫 Examiner Trap · Examiner traps

(1) In

a x + b y + cz + d = 0

the NORMAL is (a,b,c) — a plane is fixed by a point AND a normal. (2)

r \cdot \overset{n}{^} = p

needs a UNIT normal

\overset{n}{^}

, and then p is the distance from the origin. (3) Intercept form

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

gives the axis intercepts directly. (4) Plane through 3 points: set the

3 \times 3

determinant of the edge vectors to

0

Planes — Angles, Distances & Family

★ Remember · Angle & distance with planes

Angle between planes

=

angle between normals:

cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣∣ n _{2} ∣}

. Distance of a point from a plane:

\frac{∣ a x _{1} + b y _{1} + c z _{1} + d ∣}{a ^{2} + b ^{2} + c ^{2}}

Parallel & perpendicular planes

▸Parallel: $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$ (normals proportional).
▸Perpendicular: $n_{1} \cdot n_{2} = a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$ .
▸Distance between parallel planes: $\frac{∣ d _{1} - d _{2} ∣}{a ^{2} + b ^{2} + c ^{2}}$ .
▸A plane parallel to $a x + b y + cz + d = 0$ is $a x + b y + cz + k = 0$ .

🎯 Exam · Family of planes

Every plane through the line of intersection of

P_{1} = 0

and

P_{2} = 0

P_{1} + λ P_{2} = 0

. Fix

λ

from one extra condition (a point it passes through, or a required angle/parallelism).

⚠️ Watch out · Keep the same coefficients for parallel-plane distance

The formula

\frac{∣ d _{1} - d _{2} ∣}{a ^{2} + b ^{2} + c ^{2}}

needs the two planes written with

identical

a,b,c. If they aren't, scale one equation first (e.g.

2 x + 2 y = 5

x + y = 1

\Rightarrow

rewrite as

x + y = \frac{5}{2}

🚫 Examiner Trap · Examiner traps

(1) Angle between two planes

=

angle between their NORMALS; parallel

\Leftrightarrow

normals proportional, perpendicular

\Leftrightarrow n_{1} \cdot n_{2} = 0

. (2) Distance of a point:

\frac{∣ a x _{1} + b y _{1} + c z _{1} + d ∣}{a ^{2} + b ^{2} + c ^{2}}

(keep the modulus). (3) Between PARALLEL planes the numerator is

∣ d_{1} - d_{2} ∣

— but the a,b,c must be IDENTICAL first (scale one plane to match). (4) Family through the intersection of

P_{1}, P_{2}

P_{1} + λ P_{2} = 0

Line & Plane Together

★ Remember · Line-plane angle uses SINE

sin θ = \frac{∣ b \cdot n ∣}{∣ b ∣∣ n ∣}

, where

b

is the line's direction and

n

the plane's normal. It is the

sine

because the line tilts

θ

from the plane but

9 0^{\circ} - θ

from the normal.

Line-plane relationships

▸Parallel to the plane: $b \cdot n = 0$ .
▸Lies in the plane: $b \cdot n = 0$ AND a point of the line satisfies the plane.
▸Perpendicular to the plane: $b ∥ n$ .
▸Intersection: substitute the line's general point into the plane and solve for $λ$ .

🎯 Exam · Foot of perpendicular & image

Foot

from a point P to a plane: move from P along the normal

n

until you reach the plane.

Image

(reflection): go twice as far,

P^{'} = 2 (foot) - P

. For a line, drop the perpendicular using the line's direction as the constraint.

⚠️ Watch out · Don't use cosine for the line-plane angle

\frac{∣ b \cdot n ∣}{∣ b ∣∣ n ∣}

gives the angle with the

normal

— to get the angle with the plane you must take

arcsin

of it (or subtract from

9 0^{\circ}

). Writing

cos θ

here is the classic mistake.

🚫 Examiner Trap · Examiner traps

(1) Line–plane angle uses

sin θ = \frac{∣ b \cdot n ^ ∣}{∣ b ∣∣ n ^ ∣}

(SINE, not cosine — the line tilts

θ

from the plane,

9 0^{\circ} - θ

from the normal). (2) Line PARALLEL to plane

\Leftrightarrow b \cdot n = 0

; line IN the plane needs that AND a point on the plane. (3) Line

⊥

plane

\Leftrightarrow b ∥ n

. (4) Intersection: substitute the line's general point into the plane and solve for

λ

Coplanarity & Master Formula Summary

★ Remember · Coplanarity of two lines

Two lines are coplanar (a common plane exists) iff

(a_{2} - a_{1}) \cdot (b_{1} \times b_{2}) = 0

— the scalar triple product of the join and the two directions vanishes (equivalently the

3 \times 3

determinant is

0

Angle formulas at a glance

▸Line ↔ line: $cos θ = \frac{∣ b _{1} \cdot b _{2} ∣}{∣ b _{1} ∣∣ b _{2} ∣}$ .
▸Plane ↔ plane: $cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣∣ n _{2} ∣}$ .
▸Line ↔ plane: $sin θ = \frac{∣ b \cdot n ∣}{∣ b ∣∣ n ∣}$ (note the sine).
▸Point ↔ plane: $\frac{∣ a x _{1} + b y _{1} + c z _{1} + d ∣}{a ^{2} + b ^{2} + c ^{2}}$ .

🎯 Exam · Lines use directions, planes use normals

Every formula here is a dot product of the two 'characteristic vectors':

directions

b

for lines and

normals

n

for planes. The only oddity is the line-plane angle, which uses

sin

because one object is given by its direction and the other by its normal.

★ Remember · Reduce everything to point + direction/normal

Almost every 3D problem starts the same way: extract a

point

and a

direction

(line) or

normal

(plane), then apply the matching dot/cross-product formula. Set the problem up in this language and the rest is arithmetic.

🚫 Examiner Trap · Examiner traps

(1) Two lines are coplanar IFF the scalar triple product

(a_{2} - a_{1}) \cdot (b_{1} \times b_{2}) = 0

(equivalently shortest distance

= 0

). (2) Coplanar lines either INTERSECT or are PARALLEL — then a common plane exists. (3) Cartesian test: the

3 \times 3

determinant of

[(x_{2} - x_{1}, \dots), (a_{1}, b_{1}, c_{1}), (a_{2}, b_{2}, c_{2})] = 0

. (4) Use

cos

for line–line/plane–plane angles but

sin

for line–plane — the most-confused pairing.

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Frequently Asked Questions

What are the most important Three Dimensional Geometry formulas for JEE Main?

This Three Dimensional Geometry formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Coordinates of a point in space, Distance between two points, Section formula, Direction ratios and direction cosines, Angle between two intersecting lines — each shown with the key result and, where useful, a worked example.

Is this Three Dimensional Geometry formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

How should I revise Three Dimensional Geometry formulas?

Blurt the Three Dimensional Geometry formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

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