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}

Right-handed axes (x toward the viewer, y to the right, z up) with a point P(x, y, z). Dashed projection lines run from the x and y values to the foot of P in the xy-plane, then up by z to P. — A point $P (x, y, z)$ on right-handed axes, with its projections.

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

Skew lines: L1 lies on an upper plane and L2, in a different direction, on a parallel lower plane. The dotted common perpendicular d joins them vertically. — Skew lines lie in parallel planes; the common perpendicular $d$ is 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 (plane not in JEE Main 2026 syllabus)

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 (not in JEE Main 2026 syllabus)

★ 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 (not in JEE Main 2026 syllabus)

★ 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 (not in JEE Main 2026 syllabus)

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.

More JEE Main Mathematics formula sheets

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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