Math · Coordinate Geometry and Vectors

Three-dimensional Geometry formulas for JEE

Every Three-dimensional Geometry formula you need for JEE, grouped by concept.

FormulasRevision notes
32 formulas5 concepts
01Equation of a Line in 3D02Skew Lines03Equation of a Plane04Angle between Lines and Planes05Distance of a Point from a Plane
01

Equation of a Line in 3D

9 formulas

Direction Cosines from Two Points

x2−x1PQ,y2−y1PQ,z2−z1PQ\frac{x_2-x_1}{PQ}, \frac{y_2-y_1}{PQ}, \frac{z_2-z_1}{PQ}

Direction cosines of a directed line segment joining P to Q.

geometry3ddirection-cosines

Direction Cosines Identity

l2+m2+n2=1l^2 + m^2 + n^2 = 1

Sum of squares of direction cosines of a line is always 1.

applies whenl, m, n must be true direction cosines, not just direction ratios.
geometry3ddirection-cosines

Distance Between Two Points

PQ=(x2−x1)2+(y2−y1)2+(z2−z1)2PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

Spatial distance between two 3D coordinates.

geometry3ddistance

Cartesian Equation of Line (Two Points)

x−x1x2−x1=y−y1y2−y1=z−z1z2−z1\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}

Equation of a line joining two points in Cartesian form.

geometry3dlinecartesian

Vector Equation of Line (Two Points)

r⃗=a⃗+λ(b⃗−a⃗)\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})

Vector equation of a line passing through two given position vectors.

geometry3dlinevector

Parametric Equation of Line

x=x1+λa,y=y1+λb,z=z1+λcx = x_1 + \lambda a, y = y_1 + \lambda b, z = z_1 + \lambda c

Cartesian parametric coordinates of any point on a line.

geometry3dlinecartesian

Symmetric Equation of Line

x−x1a=y−y1b=z−z1c\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}

Cartesian symmetric equation of a line with direction ratios a, b, c.

applies whenCoefficients of x, y, and z must strictly be 1.
geometry3dlinecartesian

Vector Equation of Line (Point + Vector)

r⃗=a⃗+λb⃗\vec{r} = \vec{a} + \lambda \vec{b}

Vector equation of a line passing through a position vector and parallel to another vector.

geometry3dlinevector

Direction Ratios to Cosines

l=±aa2+b2+c2,m=±ba2+b2+c2,n=±ca2+b2+c2l = \pm \frac{a}{\sqrt{a^2+b^2+c^2}}, m = \pm \frac{b}{\sqrt{a^2+b^2+c^2}}, n = \pm \frac{c}{\sqrt{a^2+b^2+c^2}}

Converting direction ratios (a,b,c) to direction cosines (l,m,n).

geometry3ddirection-ratios
02

Skew Lines

5 formulas

Coplanarity of Two Lines

(a⃗2−a⃗1)⋅(b⃗1×b⃗2)=0(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0

Condition for two 3D lines to lie in the exact same plane.

geometry3dcoplanarjee-advanced

Shortest Distance (Skew Lines Vector)

d=∣(b⃗1×b⃗2)⋅(a⃗2−a⃗1)∣∣b⃗1×b⃗2∣d = \frac{|(\vec{b}_1 \times \vec{b}_2) \cdot (\vec{a}_2 - \vec{a}_1)|}{|\vec{b}_1 \times \vec{b}_2|}

Shortest perpendicular distance between two skew lines in vector form.

applies whenLines are skew (non-intersecting, non-parallel).
geometry3dskewdistance

Shortest Distance (Parallel Lines Vector)

d=∣b⃗×(a⃗2−a⃗1)∣∣b⃗∣d = \frac{|\vec{b} \times (\vec{a}_2 - \vec{a}_1)|}{|\vec{b}|}

Perpendicular distance between two parallel lines.

applies whenLines must be parallel.
geometry3dparalleldistance

Shortest Distance (Skew Cartesian)

d=∣x2−x1y2−y1z2−z1a1b1c1a2b2c2∣(b1c2−b2c1)2+(c1a2−c2a1)2+(a1b2−a2b1)2d = \frac{\left| \begin{matrix} 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{matrix} \right|}{\sqrt{(b_1c_2-b_2c_1)^2 + (c_1a_2-c_2a_1)^2 + (a_1b_2-a_2b_1)^2}}

Shortest distance between two skew lines using Cartesian determinants.

applies whenLines are skew (non-intersecting, non-parallel).
geometry3dskewdistance

Unit Normal to Skew Lines

n^=b⃗1×b⃗2∣b⃗1×b⃗2∣\hat{n} = \frac{\vec{b}_1 \times \vec{b}_2}{|\vec{b}_1 \times \vec{b}_2|}

Unit vector mutually perpendicular to two skew lines.

applies whenLines must not be parallel (cross product is non-zero).
geometry3dskewvector
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03

Equation of a Plane

6 formulas

General Equation of Plane

Ax+By+Cz+D=0Ax + By + Cz + D = 0

Cartesian general equation of a plane where A, B, C are direction ratios of the normal.

geometry3dplanejee-advanced

Family of Planes

P1+λP2=0P_1 + \lambda P_2 = 0

Equation of a plane passing through the line of intersection of two planes.

applies whenPlanes P1 and P2 must intersect.
geometry3dplanejee-advanced

Intercept Form of Plane

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

Plane equation given its intercepts a, b, c on the x, y, z axes respectively.

applies whenPlane must not pass through the origin.
geometry3dplanejee-advanced

Normal Form of Plane

r⃗⋅n^=d\vec{r} \cdot \hat{n} = d

Plane equation given its perpendicular distance from origin (d) and unit normal vector.

applies whend >= 0.
geometry3dplanejee-advanced

Point-Normal Form of Plane

(r⃗−a⃗)⋅n⃗=0(\vec{r} - \vec{a}) \cdot \vec{n} = 0

Plane passing through a position vector with a given normal vector.

geometry3dplanejee-advanced

Plane Through Three Points

[r⃗−a⃗,b⃗−a⃗,c⃗−a⃗]=0[\vec{r} - \vec{a}, \vec{b} - \vec{a}, \vec{c} - \vec{a}] = 0

Vector equation of a plane passing through three non-collinear points.

applies whenThe three points must be non-collinear.
geometry3dplanejee-advanced
04

Angle between Lines and Planes

8 formulas

Angle Between Line and Plane

sin⁡ϕ=∣b⃗⋅n⃗∣∣b⃗∣∣n⃗∣\sin\phi = \frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}||\vec{n}|}

Acute angle between a line (direction b) and a plane (normal n).

geometry3dangleplanejee-advanced

Angle Between Lines (Direction Ratios)

cos⁡θ=∣a1a2+b1b2+c1c2∣a12+b12+c12a22+b22+c22\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}}

Acute angle between two lines using their direction ratios.

geometry3dangledirection-ratios

Angle Between Lines (Direction Cosines)

cos⁡θ=∣l1l2+m1m2+n1n2∣\cos\theta = |l_1 l_2 + m_1 m_2 + n_1 n_2|

Acute angle between two lines using their direction cosines.

geometry3dangledirection-cosines

Sine Angle Between Lines (Direction Cosines)

sin⁡θ=(l1m2−l2m1)2+(m1n2−m2n1)2+(n1l2−n2l1)2\sin\theta = \sqrt{(l_1m_2-l_2m_1)^2 + (m_1n_2-m_2n_1)^2 + (n_1l_2-n_2l_1)^2}

Sine of the angle between two lines using direction cosines.

geometry3dangledirection-cosines

Sine Angle Between Lines (Direction Ratios)

sin⁡θ=(a1b2−a2b1)2+(b1c2−b2c1)2+(c1a2−c2a1)2a12+b12+c12a22+b22+c22\sin\theta = \frac{\sqrt{(a_1b_2-a_2b_1)^2 + (b_1c_2-b_2c_1)^2 + (c_1a_2-c_2a_1)^2}}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}

Sine of the angle between two lines using direction ratios.

geometry3dangledirection-ratios

Angle Between Lines (Vector Form)

cos⁡θ=∣b⃗1⋅b⃗2∣∣b⃗1∣∣b⃗2∣\cos\theta = \frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1| |\vec{b}_2|}

Acute angle between two lines given their parallel vector components.

geometry3danglevector

Condition for Parallel Lines

a1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

Condition for two lines with given direction ratios to be strictly parallel.

applies whenAngle between lines is 0 degrees.
geometry3dparallel

Condition for Perpendicular Lines

a1a2+b1b2+c1c2=0a_1a_2 + b_1b_2 + c_1c_2 = 0

Condition for two lines with given direction ratios to be strictly perpendicular.

applies whenAngle between lines is 90 degrees.
geometry3dperpendicular
05

Distance of a Point from a Plane

4 formulas

Distance Between Parallel Planes

d=∣D1−D2∣A2+B2+C2d = \frac{|D_1 - D_2|}{\sqrt{A^2+B^2+C^2}}

Shortest distance between two strictly parallel planes.

applies whenNormal direction ratios (A, B, C) must be identical for both plane equations.
geometry3ddistanceplanejee-advanced

Distance of Point from Plane

d=∣Ax1+By1+Cz1+D∣A2+B2+C2d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2+B^2+C^2}}

Perpendicular distance from a point to a plane.

geometry3ddistanceplanejee-advanced

Foot of Perpendicular on a Plane

x2−x1A=y2−y1B=z2−z1C=−Ax1+By1+Cz1+DA2+B2+C2\frac{x_2-x_1}{A} = \frac{y_2-y_1}{B} = \frac{z_2-z_1}{C} = -\frac{Ax_1+By_1+Cz_1+D}{A^2+B^2+C^2}

Coordinates of the foot of the perpendicular drawn from a point to a plane.

geometry3dfootplanejee-advanced

Image of Point in a Plane

x2−x1A=y2−y1B=z2−z1C=−2Ax1+By1+Cz1+DA2+B2+C2\frac{x_2-x_1}{A} = \frac{y_2-y_1}{B} = \frac{z_2-z_1}{C} = -2\frac{Ax_1+By_1+Cz_1+D}{A^2+B^2+C^2}

Parametric coordinates of the image of a point reflected across a plane.

geometry3dimageplanejee-advanced
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