Three Dimensional Geometry

Coordinates in Space

• A point P is represented as (x, y, z)
• x, y, z are signed distances from yz-, zx-, and xy-planes respectively
• Origin O = (0, 0, 0)

Distance Formula

• Distance between A(x₁, y₁, z₁) and B(x₂, y₂, z₂):
• AB = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
• Distance from origin to P(x, y, z): OP = √(x² + y² + z²)

Section Formula

• Internal division (m : n):
• ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n))
• External division (m : n):
• ((mx₂ − nx₁)/(m−n), (my₂ − ny₁)/(m−n), (mz₂ − nz₁)/(m−n))
• Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)

Centroid

• Triangle with vertices A, B, C:
• G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)
• Tetrahedron with vertices A, B, C, D:
• G = ((x₁+x₂+x₃+x₄)/4, (y₁+y₂+y₃+y₄)/4, (z₁+z₂+z₃+z₄)/4)

Direction Cosines & Direction Ratios

• Direction cosines: l = cosα, m = cosβ, n = cosγ
• l² + m² + n² = 1
• Direction ratios (a, b, c) are proportional to direction cosines
• Relation: l = a/√(a²+b²+c²), similarly for m, n

Angle Between Two Lines

• If direction ratios are (a₁, b₁, c₁) and (a₂, b₂, c₂):
• cosθ = (a₁a₂ + b₁b₂ + c₁c₂) / [√(a₁²+b₁²+c₁²) √(a₂²+b₂²+c₂²)]
• Perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0
• Parallel lines: a₁/a₂ = b₁/b₂ = c₁/c₂

Equation of a Straight Line

Vector Form

• r⃗ = a⃗ + λ b⃗

Symmetric Form

• (x−x₁)/l = (y−y₁)/m = (z−z₁)/n

Two-Point Form

• (x−x₁)/(x₂−x₁) = (y−y₁)/(y₂−y₁) = (z−z₁)/(z₂−z₁)

Angle Between Line and Plane

• If line has direction ratios (l, m, n) and plane is ax + by + cz + d = 0:
• sinθ = |al + bm + cn| / [√(a²+b²+c²) √(l²+m²+n²)]

Shortest Distance Between Skew Lines

• SD = |(a⃗₂ − a⃗₁) · (b⃗₁ × b⃗₂)| / |b⃗₁ × b⃗₂|

Equation of a Plane

• General form: ax + by + cz + d = 0
• Plane through origin: ax + by + cz = 0
• Intercept form: x/a + y/b + z/c = 1
• Normal form: lx + my + nz = p

Angle Between Two Planes

• cosθ = (aa′ + bb′ + cc′) / [√(a²+b²+c²) √(a′²+b′²+c′²)]

Distance of a Point from a Plane

• p = |ax₁ + by₁ + cz₁ + d| / √(a²+b²+c²)

Sphere

• Centre (a, b, c), radius r:
• (x−a)² + (y−b)² + (z−c)² = r²
• General form:
• x² + y² + z² + 2ux + 2vy + 2wz + d = 0
• Centre = (−u, −v, −w)
• Radius = √(u²+v²+w² − d)

Plane and Sphere

• If p = distance of centre from plane and r = radius:
• p < r → plane cuts sphere
• p = r → plane touches sphere
• p > r → plane does not intersect sphere

Area & Volume

• Area of triangle ABC:
• ½ |AB⃗ × AC⃗|
• Volume of tetrahedron:
• 1/6 |[AB⃗ AC⃗ AD⃗]|

JEE Main Focus Tips

• Memorise standard forms of line & plane equations
• Direction ratios are heavily tested
• Distance and angle-based questions are high frequency
• Sphere + plane combinations are scoring