Three-dimensional Coordinate Geometry
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
CrackIt
Coordinates, Distance & Section Formula
Distance Formula
For P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
PQ = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
PQ = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
Section Formula (Internal)
For ratio m : n
x = (mx₂+nx₁)/(m+n)
y = (my₂+ny₁)/(m+n)
z = (mz₂+nz₁)/(m+n)
x = (mx₂+nx₁)/(m+n)
y = (my₂+ny₁)/(m+n)
z = (mz₂+nz₁)/(m+n)
Section Formula (External)
For ratio m : n
x = (mx₂-nx₁)/(m-n)
y = (my₂-ny₁)/(m-n)
z = (mz₂-nz₁)/(m-n)
x = (mx₂-nx₁)/(m-n)
y = (my₂-ny₁)/(m-n)
z = (mz₂-nz₁)/(m-n)
Mid-Point Formula
Ratio is 1:1
P = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
P = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
Centroid of Triangle
For vertices A(x₁,y₁,z₁), B(x₂,y₂,z₂), C(x₃,y₃,z₃):
G = ((x₁+x₂+x₃)/3, ... , ...)
G = ((x₁+x₂+x₃)/3, ... , ...)
Direction Cosines (DCs) & Ratios (DRs)
Direction Cosines (l, m, n)
l = cos α, m = cos β, n = cos γ
Key Identity: l² + m² + n² = 1
Key Identity: l² + m² + n² = 1
α, β, γ are angles the line makes with the positive x, y, z axes respectively.
From DRs (a,b,c) to DCs (l,m,n)
l = ± a/√(a²+b²+c²)
m = ± b/√(a²+b²+c²)
n = ± c/√(a²+b²+c²)
m = ± b/√(a²+b²+c²)
n = ± c/√(a²+b²+c²)
DRs are any set of numbers proportional to the DCs.
DRs of Line joining P & Q
For P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
DRs are (x₂-x₁, y₂-y₁, z₂-z₁)
DRs are (x₂-x₁, y₂-y₁, z₂-z₁)
Angle between Two Lines
Using DRs (a₁,b₁,c₁) & (a₂,b₂,c₂):
cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(Σa₁²) √(Σa₂²))
Also, cos θ = |l₁l₂ + m₁m₂ + n₁n₂|
cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(Σa₁²) √(Σa₂²))
Also, cos θ = |l₁l₂ + m₁m₂ + n₁n₂|
The Straight Line in Space
Vector & Cartesian Equations
Vector: r = a + λb
Cartesian: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
Cartesian: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
Line passes through point 'a' (x₁,y₁,z₁) and is parallel to vector 'b' with DRs (a,b,c).
Line Through Two Points
Vector: r = a + λ(b-a)
Cartesian: (x-x₁)/(x₂-x₁) = (y-y₁)/(y₂-y₁) = (z-z₁)/(z₂-z₁)
Cartesian: (x-x₁)/(x₂-x₁) = (y-y₁)/(y₂-y₁) = (z-z₁)/(z₂-z₁)
Line passes through points with position vectors 'a' and 'b'.
Skew Lines & Shortest Distance
Shortest Distance (Skew Lines)
For r = a₁ + λb₁ and r = a₂ + μb₂:
SD = | (b₁ × b₂) ⋅ (a₂ - a₁) | / | b₁ × b₂ |
SD = | (b₁ × b₂) ⋅ (a₂ - a₁) | / | b₁ × b₂ |
If SD = 0, the lines intersect. The numerator is a scalar triple product.
Distance (Parallel Lines)
For r = a₁ + λb and r = a₂ + μb:
D = | b × (a₂ - a₁) | / | b |
D = | b × (a₂ - a₁) | / | b |
The direction vectors 'b' are the same for parallel lines.
The Plane
General & Normal Forms
General: ax + by + cz + d = 0
Vector Normal: r ⋅ n̂ = p
Cartesian Normal: lx + my + nz = p
Vector Normal: r ⋅ n̂ = p
Cartesian Normal: lx + my + nz = p
(a,b,c) are DRs of the normal. 'p' is the perpendicular distance from the origin.
Intercept Form
x/a + y/b + z/c = 1
Plane makes intercepts a, b, c on the x, y, z axes respectively.
Plane through Intersection
For planes P₁=0 and P₂=0:
P₁ + λP₂ = 0
P₁ + λP₂ = 0
This represents a family of planes passing through the line of intersection of P₁ and P₂.
Angles & Distances involving Planes
Angle between Line & Plane
Line: r = a+λb, Plane: r⋅n=d
sin θ = | b ⋅ n | / (|b| |n|)
sin θ = | b ⋅ n | / (|b| |n|)
This is the ONLY case with sin θ. All others use cos θ. A very common trap!
Angle between Two Planes
Planes: r⋅n₁=d₁ and r⋅n₂=d₂
cos θ = | n₁ ⋅ n₂ | / (|n₁| |n₂|)
cos θ = | n₁ ⋅ n₂ | / (|n₁| |n₂|)
Distance of Point from Plane
Point (x₁,y₁,z₁) & Plane ax+by+cz+d=0:
D = |ax₁ + by₁ + cz₁ + d| / √(a²+b²+c²)
D = |ax₁ + by₁ + cz₁ + d| / √(a²+b²+c²)
Distance between Parallel Planes
Planes ax+by+cz+d₁=0 & ax+by+cz+d₂=0:
D = |d₁ - d₂| / √(a²+b²+c²)
D = |d₁ - d₂| / √(a²+b²+c²)
Speed Hacks for 3D Geometry
Vector form is often faster for shortest distance and angle problems. Convert Cartesian to vector if needed.
For 'Image' or 'Foot of Perpendicular' questions, memorize the direct formula. Deriving it is too slow for BITSAT.
Check for parallel/perpendicular conditions first (a₁a₂+b₁b₂+c₁c₂=0 or a₁/a₂=b₁/b₂=c₁/c₂). It can simplify the problem immensely.
Use options to verify. If a question asks for a point on a line/plane, plug the options back into the equation.
Remember sin θ for line-plane angle and cos θ for line-line or plane-plane angles. This is a common trap.
Line vs. Plane: Quick Comparison
| Property | Line | Plane |
|---|---|---|
Defining Elements | A point & a parallel vector | A point & a normal vector |
General Cartesian Eq. | (x-x₁)/a = (y-y₁)/b = (z-z₁)/c | ax + by + cz + d = 0 |
Direction Vectors (a,b,c) | DRs are PARALLEL to the line | DRs are NORMAL (perpendicular) to the plane |
Angle Convention | cos θ with another line | cos θ with another plane |
Mixed Angle | sin θ with a plane | sin θ with a line |