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MathematicsVectors & 3D GeometryTop Priority (3-4 Qs in JEE)

Three Dimensional Geometry (3D) Formulas

Direction cosines & ratios, equations of lines in 3D, shortest distance between skew and parallel lines.

9 Formulas•1,174+ Questions in Bank

Concept Diagram

Vector Cross Product & Right-Hand Rule
a⃗b⃗a⃗ × b⃗ (⊥ to plane)θ|a⃗ × b⃗| = |a||b| sin θa⃗ · b⃗ = |a||b| cos θ
Vector product a × b yielding perpendicular vector |a||b|sin(θ)n̂ and parallelogram area.

Core Formulas & Mathematical Identities

9
01

Direction Cosines & Direction Ratios

l=cos⁡α,  m=cos⁡β,  n=cos⁡γ  ⟹  l2+m2+n2=1,l=aa2+b2+c2l = \cos\alpha, \; m = \cos\beta, \; n = \cos\gamma \implies l^2 + m^2 + n^2 = 1, \quad l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}

Direction cosines represent components of the unit direction vector in 3D.

Examiner Tip: cos²α + cos²β + cos²γ = 1 implies sin²α + sin²β + sin²γ = 2.
02

Equation of a Line in 3D

x−x1a=y−y1b=z−z1c=λ  ⟺  r⃗=a⃗+λb⃗\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} = \lambda \iff \vec{r} = \vec{a} + \lambda \vec{b}

Symmetric and vector forms of line passing through (x₁, y₁, z₁) with direction ratios <a, b, c>.

Examiner Tip: Any general point on the line is (x₁ + aλ, y₁ + bλ, z₁ + cλ).
03

Shortest Distance Between Skew Lines

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

Minimum distance between non-parallel, non-intersecting skew lines in 3D.

Examiner Tip: Lines intersect if and only if (a₂ - a₁) · (b₁ × b₂) = 0.
04

Shortest Distance Between Parallel Lines

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

Distance between parallel lines r = a₁ + λb and r = a₂ + μb.

Examiner Tip: Uses cross product magnitude divided by |b|.
05

Angle Between Two Lines in 3D

cos⁡θ=∣a1a2+b1b2+c1c2∣a12+b12+c12a22+b22+c22=∣b⃗1⋅b⃗2∣∣b⃗1∣∣b⃗2∣\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}} = \frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}

Acute angle between lines with direction ratios.

Examiner Tip: Lines are perpendicular if a₁a₂ + b₁b₂ + c₁c₂ = 0.
06

Condition for Coplanarity of Two Lines

∣x2−x1y2−y1z2−z1a1b1c1a2b2c2∣=0  ⟺  (a⃗2−a⃗1)⋅(b⃗1×b⃗2)=0\begin{vmatrix} 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{vmatrix} = 0 \iff (\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0

Criterion for two straight lines in 3D space to lie in the same unique plane.

Examiner Tip: If coplanar, either the lines intersect at a point or they are parallel.
07

Distance of a Point from a Line in 3D

d=∣(p⃗−a⃗)×b⃗∣∣b⃗∣=∣p⃗−a⃗∣2−((p⃗−a⃗)⋅b⃗∣b⃗∣)2d = \frac{|(\vec{p} - \vec{a}) \times \vec{b}|}{|\vec{b}|} = \sqrt{|\vec{p} - \vec{a}|^2 - \left(\frac{(\vec{p} - \vec{a}) \cdot \vec{b}}{|\vec{b}|}\right)^2}

Perpendicular distance from point P(p) to line r = a + λb.

Examiner Tip: Foot of perpendicular F: let F = a + λb, then (p - F) · b = 0 determines λ directly.
08

Equation of Sphere & Its Properties

x2+y2+z2+2ux+2vy+2wz+d=0  ⟹  Centre=(−u,−v,−w),  r=u2+v2+w2−dx^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 \implies \text{Centre} = (-u,-v,-w), \; r = \sqrt{u^2 + v^2 + w^2 - d}

General second-degree equation of a sphere. Standard form with centre (a,b,c): (x-a)² + (y-b)² + (z-c)² = r².

Examiner Tip: Sphere through origin: d = 0. Diameter-form sphere with endpoints (x₁,y₁,z₁) and (x₂,y₂,z₂): (x-x₁)(x-x₂) + (y-y₁)(y-y₂) + (z-z₁)(z-z₂) = 0. For a point to lie on sphere: substitute and check if the equation is satisfied.
09

Image of a Point in a Plane & Foot of Perpendicular

x′−x1a=y′−y1b=z′−z1c=−2⋅ax1+by1+cz1+da2+b2+c2\frac{x' - x_1}{a} = \frac{y' - y_1}{b} = \frac{z' - z_1}{c} = -2\cdot\frac{ax_1 + by_1 + cz_1 + d}{a^2 + b^2 + c^2}

Coordinates of image (x', y', z') of point (x₁, y₁, z₁) in plane ax + by + cz + d = 0. Foot of perpendicular F has parameter k = −(ax₁+by₁+cz₁+d)/(a²+b²+c²).

Examiner Tip: F (foot of ⊥) = (x₁ + ak, y₁ + bk, z₁ + ck). Image point = (x₁ + 2ak, y₁ + 2bk, z₁ + 2ck). Distance from point to plane: |ax₁+by₁+cz₁+d| / √(a²+b²+c²).

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Three Dimensional Geometry (3D)

  • •Direction cosines satisfy l² + m² + n² = 1, whereas direction ratios a, b, c do not have to sum to 1.
  • •Shortest distance between skew lines formula requires the scalar triple product with unit vector in denominator.
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