Math · Coordinate Geometry and Vectors

Introduction to Three-dimensional Geometry formulas for JEE

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

FormulasRevision notes
9 formulas2 concepts
01Introduction to 3D Geometry02Distance and Section Formulae in 3D
01

Introduction to 3D Geometry

1 formula

Direction Cosines Identity

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

The sum of the squares of the direction cosines of a line is always 1.

direction-cosinesidentityjee-advanced
02

Distance and Section Formulae in 3D

8 formulas

Centroid of a Tetrahedron

G=(x1+x2+x3+x44,y1+y2+y3+y44,z1+z2+z3+z44)G = \left( \frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}, \frac{z_1+z_2+z_3+z_4}{4} \right)

Coordinates of the centroid of a tetrahedron given its four vertices.

centroidtetrahedronjee-advanced

Centroid of a Triangle

G=(x1+x2+x33,y1+y2+y33,z1+z2+z33)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)

Coordinates of the centroid of a triangle given its three vertices.

centroidtriangle

Collinearity using distance

AB+BC=ACAB + BC = AC

Condition for three points A, B, and C to be collinear using the distance formula.

applies whenPoints must be ordered A, B, then C on the line.
collineardistance

Distance between two points

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

Calculates the Euclidean distance between two points in 3D space.

distance3d-geometry

Distance from origin

d=x2+y2+z2d = \sqrt{x^2 + y^2 + z^2}

Calculates the distance of a point from the origin (0,0,0).

applies whenOrigin is strictly (0,0,0)
distanceoriginspecial-case

Midpoint Formula

M=(x1+x22,y1+y22,z1+z22)M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right)

Coordinates of the midpoint of a line segment.

midpointjee-advanced

Section Formula (External)

P=(mx2−nx1m−n,my2−ny1m−n,mz2−nz1m−n)P = \left( \frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n}, \frac{mz_2-nz_1}{m-n} \right)

Coordinates of a point dividing a line segment joining two points externally in the ratio m:n.

applies whenm≠nm \neq n
sectionexternaljee-advanced

Section Formula (Internal)

P=(mx2+nx1m+n,my2+ny1m+n,mz2+nz1m+n)P = \left( \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n} \right)

Coordinates of a point dividing a line segment joining two points internally in the ratio m:n.

applies whenm+n≠0m+n \neq 0
sectioninternaljee-advanced
Other chapters
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