Distance & Section Formula
Distance = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
Section Formula (m:n) = ( (mx₂ + nx₁)/(m+n) , (my₂ + ny₁)/(m+n) , (mz₂ + nz₁)/(m+n) )
Internal & External division both covered
Direction Cosines & Direction Ratios
Direction Cosines
l = cosα, m = cosβ, n = cosγ
l² + m² + n² = 1
Direction Ratios
a, b, c such that l = a/√(a²+b²+c²), etc.
Two vectors parallel: a₁/a₂ = b₁/b₂ = c₁/c₂
Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0
Equation of Line in 3D
| Type | Vector Form | Cartesian (Symmetric) Form |
|---|---|---|
| Through point A(a) parallel to b | r = a + λb | (x−x₁)/l = (y−y₁)/m = (z−z₁)/n |
| Through two points A(a), B(b) | r = a + λ(b − a) | (x−x₁)/(x₂−x₁) = (y−y₁)/(y₂−y₁) = (z−z₁)/(z₂−z₁) |
Angle Between Two Lines
cosθ = |l₁l₂ + m₁m₂ + n₁n₂|
Lines parallel if direction ratios proportional
Lines perpendicular if l₁l₂ + m₁m₂ + n₁n₂ = 0
Shortest Distance Between Skew Lines
Distance = |(r₂ − r₁) · (d₁ × d₂)| / |d₁ × d₂|
For lines r = a₁ + λb₁ and r = a₂ + μb₂
Equation of Plane
| Form | Equation |
|---|---|
| General | ax + by + cz + d = 0 |
| Normal form | lx + my + nz = p |
| Through point (x₁,y₁,z₁) | a(x−x₁) + b(y−y₁) + c(z−z₁) = 0 |
| Through three points | Determinant form = 0 |
| Intercept form | x/a + y/b + z/c = 1 |
Angle Between Plane & Line
sinφ = |l a + m b + n c| / (√(l²+m²+n²) √(a²+b²+c²))
φ = angle between line & normal → actual angle = 90° − φ
JEE Key Points
Must master: Vector & Cartesian forms of line, shortest distance formula, plane in all forms
Frequent questions: Angle between lines/planes, coplanarity of lines, distance from point to plane, image of point in plane
Weightage: 3–5 questions (10–15 marks) every year