Mathematics · JEE

Three Dimensional Geometry Formula Sheet for JEE

36+ JEE formulas in this unit

Quick answer

The Three Dimensional Geometry JEE formula sheet lists 36+ important formulas for JEE Main and Advanced, including essential identities from Coordinates of a point in space, the distance between two points, section formula, Direction ratios and direction cosines and the angle between two intersecting lines, Equation of a line; Skew lines, the shortest distance between them and its equation. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Three Dimensional Geometry. This unit-wise formula list covers 36+ exam-relevant results across Coordinates of a point in space, the distance between two points, section formula, Direction ratios and direction cosines and the angle between two intersecting lines, Equation of a line; Skew lines, the shortest distance between them and its equation, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

36 formulas across 3 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 22Important: 12Supplementary: 2

Coordinates of a point in space, the distance between two points, section formula

Coordinates of a point in spaceEssential

P(x,y,z)

Variables: $x,y,z$ are the Cartesian coordinates of point $P$ measured along mutually perpendicular axes.
Conditions: Three-dimensional Cartesian coordinate system.
Where used in JEE: Basic representation of points in 3D geometry problems.

Distance of a point from the originEssential

OP=\sqrt{x^2+y^2+z^2}

Variables: $P(x,y,z)$ , $O(0,0,0)$ .
Where used in JEE: Finding position vector magnitude, sphere/radius type problems, coordinate computations.

Distance between two points in spaceEssential

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

Variables: $P(x_1,y_1,z_1),\ Q(x_2,y_2,z_2)$ .
Where used in JEE: Lengths, triangle geometry in 3D, collinearity checks, locus and section problems.

Square of distance between two pointsImportant

PQ^2=(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2

Variables: $P(x_1,y_1,z_1),\ Q(x_2,y_2,z_2)$ .
Where used in JEE: Avoiding square roots in equality/comparison problems, proving perpendicularity or midpoint relations.

Coordinates of midpoint of a line segmentEssential

M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)

Variables: $M$ is midpoint of segment joining $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ .
Where used in JEE: Midpoint, centroid, symmetry and line-equation problems.

Internal section formula in 3DEssential

R\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n}\right)

Variables: $R$ divides the segment joining $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ internally in the ratio $m:n$ , i.e. $PR:RQ=m:n$ .
Conditions: $m,n>0$ , $m+n\neq 0$ .
Where used in JEE: Finding points dividing a segment in a given ratio, medians and centroid type problems.

External section formula in 3DImportant

R\left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n},\frac{mz_2-nz_1}{m-n}\right)

Variables: $R$ divides the line joining $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ externally in the ratio $m:n$ .
Conditions: $m,n>0$ , $m\neq n$ .
Where used in JEE: External division and harmonic division type coordinate problems.

Centroid of a triangle in spaceImportant

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)

Variables: Vertices are $A(x_1,y_1,z_1), B(x_2,y_2,z_2), C(x_3,y_3,z_3)$ .
Where used in JEE: Coordinate geometry of triangles in space; median and centroid questions.

Direction ratios and direction cosines and the angle between two intersecting lines

Direction ratios of a lineEssential

(a,b,c)\propto

any vector parallel to the line

Variables: $a,b,c$ are direction ratios (d.r.s) of the line.
Conditions: Not all of $a,b,c$ are zero.
Where used in JEE: Writing equations of lines, testing parallelism/perpendicularity, angle problems.

Direction cosines of a lineEssential

l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma

Variables: $\alpha,\beta,\gamma$ are the angles made by the line with positive $x,y,z$ -axes respectively; $l,m,n$ are direction cosines (d.c.s).
Where used in JEE: Finding orientation of a line, angle relations with axes.

Fundamental relation among direction cosinesEssential

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

Variables: $l,m,n$ are direction cosines of a line.
Where used in JEE: Verification of d.c.s, recovering one cosine from the other two.

Direction cosines from direction ratiosEssential

l=\pm\frac{a}{\sqrt{a^2+b^2+c^2}},\quad m=\pm\frac{b}{\sqrt{a^2+b^2+c^2}},\quad n=\pm\frac{c}{\sqrt{a^2+b^2+c^2}}

Variables: $(a,b,c)$ are direction ratios; $(l,m,n)$ are corresponding direction cosines.
Conditions: Signs are taken consistently so that $(l:m:n)=(a:b:c)$ .
Where used in JEE: Converting d.r.s to unit direction data; line-angle questions.

Direction ratios from direction cosinesImportant

a:b:c=l:m:n

Variables: $(a,b,c)$ are direction ratios; $(l,m,n)$ are direction cosines.
Conditions: Any nonzero proportional triple may be taken as d.r.s.
Where used in JEE: Obtaining line equation from orientation information.

Line through two given points: direction ratiosEssential

(a,b,c)=(x_2-x_1,\ y_2-y_1,\ z_2-z_1)

Variables: Line passes through $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ .
Conditions: $P\neq Q$ .
Where used in JEE: Writing equation of a line through two points; checking collinearity.

Angle between two lines using direction cosinesEssential

\cos\theta=l_1l_2+m_1m_2+n_1n_2

Variables: $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2)$ are direction cosines of the two lines; $\theta$ is the angle between them.
Conditions: $0\leq \theta\leq \pi$ .
Where used in JEE: Finding acute/obtuse angle between intersecting lines.

Angle between two lines using direction ratiosEssential

\cos\theta=\frac{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}}

Variables: $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$ are direction ratios of the two lines.
Where used in JEE: Most common JEE formula for angle between two lines.

Condition for perpendicular linesEssential

a_1a_2+b_1b_2+c_1c_2=0

Variables: $(a_1,b_1,c_1)$ , $(a_2,b_2,c_2)$ are direction ratios of the two lines.
Conditions: Both direction-ratio triples are nonzero.
Where used in JEE: Testing orthogonality of lines in 3D.

Condition for parallel linesEssential

\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}

Variables: $(a_1,b_1,c_1)$ , $(a_2,b_2,c_2)$ are direction ratios of the two lines.
Conditions: Ratios are interpreted where denominators are nonzero; equivalently the two triples are proportional.
Where used in JEE: Checking parallelism; identifying skew/non-skew cases.

Angles of a line with the coordinate axesImportant

\cos\alpha=l,\ \cos\beta=m,\ \cos\gamma=n

Variables: $\alpha,\beta,\gamma$ are inclinations of the line with positive axes; $l,m,n$ are d.c.s.
Where used in JEE: Converting axis-angle information into d.c.s or vice versa.

Equation of a line; Skew lines, the shortest distance between them and its equation

Cartesian equation of a line through a point and with direction ratiosEssential

\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}

Variables: $(x_1,y_1,z_1)$ is a point on the line and $(a,b,c)$ are direction ratios.
Conditions: Use suitable reduced form if any of $a,b,c$ is zero.
Where used in JEE: Standard equation of line in symmetric form.

Parametric equation of a line through a pointEssential

x=x_1+a\lambda,\quad y=y_1+b\lambda,\quad z=z_1+c\lambda

Variables: $(x_1,y_1,z_1)$ is a point on the line; $(a,b,c)$ are direction ratios; $\lambda\in\mathbb{R}$ is parameter.
Where used in JEE: Intersection, shortest distance, image of moving point, elimination of parameter.

Vector form of a lineImportant

\vec r=\vec a+\lambda\vec b

Variables: $\vec a$ is position vector of a fixed point on the line, $\vec b$ is a vector parallel to the line, $\lambda\in\mathbb{R}$ .
Conditions: $\vec b\neq \vec 0$ .
Where used in JEE: Compact representation of lines; vector methods in shortest distance problems.

Line through two given points: parametric formEssential

x=x_1+\lambda(x_2-x_1),\quad y=y_1+\lambda(y_2-y_1),\quad z=z_1+\lambda(z_2-z_1)

Variables: The line passes through $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ .
Conditions: $P\neq Q$ .
Where used in JEE: Constructing line equation from endpoints.

Line through two given points: symmetric formImportant

\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}

Variables: Line through $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ .
Conditions: Use modified form if any denominator is zero.
Where used in JEE: Direct line equation from two points.

Condition that a point lies on a line in symmetric formImportant

\frac{x_0-x_1}{a}=\frac{y_0-y_1}{b}=\frac{z_0-z_1}{c}

Variables: $(x_0,y_0,z_0)$ is the test point; line is $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$ .
Conditions: Interpret with care if any of $a,b,c$ is zero.
Where used in JEE: Point-line incidence and collinearity checks.

Coplanarity condition for two linesEssential

\begin{vmatrix}x_2-x_1 & y_2-y_1 & z_2-z_1\\ a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\end{vmatrix}=0

Variables: Lines are $\vec r=\vec r_1+\lambda(a_1,b_1,c_1)$ and $\vec r=\vec r_2+\mu(a_2,b_2,c_2)$ ; points $(x_1,y_1,z_1)$ , $(x_2,y_2,z_2)$ lie on them respectively.
Where used in JEE: Testing whether two non-parallel lines intersect or are skew.

Condition for skew linesImportant

\begin{vmatrix}x_2-x_1 & y_2-y_1 & z_2-z_1\\ a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\end{vmatrix}\neq 0

Variables: Lines have points $(x_1,y_1,z_1)$ , $(x_2,y_2,z_2)$ and direction ratios $(a_1,b_1,c_1)$ , $(a_2,b_2,c_2)$ .
Conditions: Lines are non-parallel.
Where used in JEE: Identifying skew lines before applying shortest distance formula.

Shortest distance between two skew lines using scalar triple productEssential

d=\frac{|(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}

Variables: Lines are $\vec r=\vec a_1+\lambda\vec b_1$ and $\vec r=\vec a_2+\mu\vec b_2$ .
Conditions: Applicable for non-parallel lines, so $\vec b_1\times\vec b_2\neq \vec 0$ .
Where used in JEE: Standard JEE formula for shortest distance between skew lines.

Shortest distance between two skew lines in determinant formEssential

d=\frac{\left|\begin{vmatrix}x_2-x_1 & y_2-y_1 & z_2-z_1\\ a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\end{vmatrix}\right|}{\sqrt{(b_1c_2-b_2c_1)^2+(c_1a_2-c_2a_1)^2+(a_1b_2-a_2b_1)^2}}

Variables: Lines pass through $(x_1,y_1,z_1)$ , $(x_2,y_2,z_2)$ and have direction ratios $(a_1,b_1,c_1)$ , $(a_2,b_2,c_2)$ .
Conditions: Lines are non-parallel.
Where used in JEE: Coordinate form of shortest distance between skew lines.

Shortest distance between parallel linesImportant

d=\frac{|(\vec a_2-\vec a_1)\times\vec b|}{|\vec b|}

Variables: Parallel lines are $\vec r=\vec a_1+\lambda\vec b$ and $\vec r=\vec a_2+\mu\vec b$ .
Conditions: $\vec b\neq \vec 0$ .
Where used in JEE: Distance between two parallel lines in space.

Shortest distance between parallel lines in direction-ratio formSupplementary

d=\frac{\sqrt{|\vec{PQ}|^2(a^2+b^2+c^2)-\big((x_2-x_1)a+(y_2-y_1)b+(z_2-z_1)c\big)^2}}{\sqrt{a^2+b^2+c^2}}

Variables: Parallel lines have common direction ratios $(a,b,c)$ ; $P(x_1,y_1,z_1)$ , $Q(x_2,y_2,z_2)$ are points on them; $|\vec{PQ}|^2=(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2$ .
Where used in JEE: Coordinate computation of distance between parallel lines.

Equation of line of shortest distance between two skew lines: direction vectorEssential

\vec d=\vec b_1\times\vec b_2

Variables: $\vec b_1,\vec b_2$ are direction vectors of the two skew lines; $\vec d$ is direction vector of their common perpendicular.
Conditions: $\vec b_1\times\vec b_2\neq \vec 0$ .
Where used in JEE: Constructing the line representing the shortest distance between skew lines.

Condition for feet of common perpendicular on two skew linesEssential

(\vec r_2-\vec r_1)\cdot\vec b_1=0,\quad (\vec r_2-\vec r_1)\cdot\vec b_2=0

Variables: $\vec r_1=\vec a_1+\lambda\vec b_1$ and $\vec r_2=\vec a_2+\mu\vec b_2$ are variable points on the two skew lines.
Conditions: Used to determine parameters $\lambda,\mu$ of the feet of the common perpendicular.
Where used in JEE: Finding actual endpoints of shortest distance segment and then its equation.

Equation of the common perpendicular through one foot pointImportant

\vec r=\vec r_1+t(\vec b_1\times\vec b_2)

Variables: $\vec r_1$ is one foot of the common perpendicular on the first line; $t\in\mathbb{R}$ .
Conditions: $\vec r_1$ must satisfy the foot conditions with some point on the second line.
Where used in JEE: Writing the equation of the shortest-distance line once a foot point is known.

Equivalent equation of common perpendicular through the other foot pointSupplementary

\vec r=\vec r_2+t(\vec b_1\times\vec b_2)

Variables: $\vec r_2$ is the foot of the common perpendicular on the second line.
Conditions: $\vec r_2$ corresponds to the same common perpendicular.
Where used in JEE: Alternative form of the shortest-distance line.

Intersecting lines condition via parametersImportant

x_1+a_1\lambda=x_2+a_2\mu,\quad y_1+b_1\lambda=y_2+b_2\mu,\quad z_1+c_1\lambda=z_2+c_2\mu

Variables: Two lines are $x=x_1+a_1\lambda, y=y_1+b_1\lambda, z=z_1+c_1\lambda$ and $x=x_2+a_2\mu, y=y_2+b_2\mu, z=z_2+c_2\mu$ .
Conditions: A common solution $(\lambda,\mu)$ implies intersection.
Where used in JEE: Testing whether two lines intersect and finding point of intersection.

Frequently asked questions

What are the important Three Dimensional Geometry formulas for JEE?

This page lists 36+ JEE-relevant Three Dimensional Geometry formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Three Dimensional Geometry formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Three Dimensional Geometry, covering Coordinates of a point in space, the distance between two points, section formula, Direction ratios and direction cosines and the angle between two intersecting lines, Equation of a line; Skew lines, the shortest distance between them and its equation.

How should I revise the Three Dimensional Geometry formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Three Dimensional Geometry MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Three Dimensional Geometry?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.