Three-Dimensional Geometry formulas for JEE Main & Advanced

24 formula entries for Three-Dimensional Geometry, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Direction Cosines (DC) of a Line

l = \cos α, m = \cos β, n = \cos γ

When to use. Use when you need the actual angles a line makes with axes or to find the line's orientation uniquely.

Trap. Direction cosines are always for a directed line; reversing the direction changes signs of all three.

Direction Ratios (DR) of a Line

DRs: a, b, c \Rightarrow (l, m, n) = \pm \frac{(a, b, c)}{\sqrt{a^{2} + b^{2} + c^{2}}}

(one common sign for all three)

When to use. Use when you have a vector along the line (e.g., from two points) and need to find DCs or check parallelism.

Trap. Direction ratios are not unique; any scalar multiple gives the same line.

Angle Between Two Lines (using DCs/DRs)

\cos θ = ∣ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} ∣ = \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}}}

When to use. Use to find the angle between any two lines given their DCs or DRs.

Trap. For acute angle, take absolute value; for obtuse, use negative sign.

Condition for Perpendicular and Parallel Lines

Perpendicular: a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0 Parallel: \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}

When to use. Use to quickly check geometric relationships between lines in 3D.

Trap. For perpendicular, use the dot product of DRs (not DCs) directly; no absolute value needed.

Equation of a Line in Space: Vector Form

\vec{r} = \vec{a} + λ \vec{b}

When to use. Use when you know a point on the line and its direction (e.g., from a parallel vector).

Trap. The direction vector b must be non-zero; λ is any real number.

Equation of a Line in Space: Cartesian Form (Symmetric)

\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c} = λ

When to use. Use when coordinates of a point and direction ratios are known; common in JEE problems.

Trap. If any DR is zero (e.g., a=0), the equation becomes x = x₁ (constant), and the other ratios still hold.

Equation of Line Through Two Points

\frac{x - x_{1}}{x_{2} - x_{1}} = \frac{y - y_{1}}{y_{2} - y_{1}} = \frac{z - z_{1}}{z_{2} - z_{1}}

When to use. Use when two points on the line are given.

Trap. If any coordinate difference is zero, treat that numerator as zero (e.g., x = x₁).

Equation of a Plane: General Form

a x + b y + c z + d = 0

When to use. Use as the standard form to represent any plane; often used to find distance from a point.

Trap. The coefficients a,b,c are not unique; multiplying by a constant gives the same plane.

Equation of a Plane: Intercept Form

\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1

When to use. Use when the intercepts on axes are given or easily found.

Trap. Intercepts can be negative if the plane cuts the negative axes.

Equation of a Plane: Normal Form

l x + m y + n z = p

When to use. Use to find the perpendicular distance from origin to a plane.

Trap. The normal vector must be a unit vector; otherwise, divide by its magnitude.

Angle Between Two Planes

\cos θ = \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}}}

When to use. Use to find the dihedral angle between two planes.

Trap. Take absolute value for acute angle; for obtuse, use negative.

Condition for Perpendicular and Parallel Planes

Perpendicular: a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0 Parallel: \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}

When to use. Use to check geometric relationships between planes.

Trap. For parallel planes, the constant term d may differ; that gives distinct parallel planes.

Distance of a Point from a Plane

Distance = \frac{∣ a x_{1} + b y_{1} + c z_{1} + d ∣}{\sqrt{a^{2} + b^{2} + c^{2}}}

When to use. Use to find the shortest distance from a point to a plane.

Trap. Use absolute value; distance is always non-negative.

Distance Between Two Parallel Planes

Distance = \frac{∣ d_{1} - d_{2} ∣}{\sqrt{a^{2} + b^{2} + c^{2}}}

When to use. Use when given two parallel planes.

Trap. The planes must have the same normal coefficients (a,b,c); otherwise, they are not parallel.

Equation of a Plane Through a Given Point and Parallel to Two Lines

∣ \begin{array}{ccc} x - x_{1} & y - y_{1} & z - z_{1} \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array} ∣ = 0

When to use. Use when a plane is defined by a point and two direction vectors (e.g., containing two lines).

Trap. The two lines must not be parallel; otherwise, the plane is not uniquely defined.

Equation of a Plane Through Three Points

∣ \begin{array}{ccc} x - x_{1} & y - y_{1} & z - z_{1} \\ x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\ x_{3} - x_{1} & y_{3} - y_{1} & z_{3} - z_{1} \end{array} ∣ = 0

When to use. Use when three points on the plane are given.

Trap. Points must be non-collinear; otherwise, the determinant is zero for all (x,y,z).

Angle Between a Line and a Plane

\sin ϕ = \frac{∣ a A + b B + c C ∣}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{A^{2} + B^{2} + C^{2}}}

When to use. Use to find the angle at which a line meets a plane.

Trap. Use sin, not cos; the angle between line and plane is complementary to the angle between line and normal.

Condition for a Line to Lie in a Plane

Line: \frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}, Plane: A x + B y + C z + D = 0 Conditions: a A + b B + c C = 0 and A x_{1} + B y_{1} + C z_{1} + D = 0

When to use. Use to check if a given line is contained in a plane.

Trap. Both conditions must hold; one alone is insufficient.

Shortest Distance Between Two Skew Lines

SD = \frac{∣ ({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2}) ∣}{∣ {\vec{b}}_{1} \times {\vec{b}}_{2} ∣}

When to use. Use to find the shortest distance between two lines that are not parallel and do not intersect.

Trap. Lines must be skew (non-parallel, non-intersecting); if parallel, use distance between parallel lines formula.

Shortest Distance Between Two Parallel Lines

Distance = \frac{∣ ({\vec{a}}_{2} - {\vec{a}}_{1}) \times \vec{b} ∣}{∣ \vec{b} ∣}

When to use. Use when two lines are parallel (DRs proportional).

Trap. The direction vector b must be the same (or proportional) for both lines.

Intersection of a Line and a Plane

Line: \frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c} = λ Plane: A x + B y + C z + D = 0 \Rightarrow A (x_{1} + λ a) + B (y_{1} + λ b) + C (z_{1} + λ c) + D = 0

When to use. Use to find the point where a line meets a plane.

Trap. If the line is parallel to the plane (aA+bB+cC=0), there is no intersection (or infinite if line lies in plane).

Foot of Perpendicular from a Point to a Plane

t = \frac{a x_{1} + b y_{1} + c z_{1} + d}{a^{2} + b^{2} + c^{2}} Foot: (x_{1} - a t, y_{1} - b t, z_{1} - c t)

When to use. Use to find the projection of a point onto a plane.

Trap. The sign of t depends on the plane equation; ensure correct sign by checking that the foot lies on the plane.

Image of a Point in a Plane

Image: (x_{1} - 2 a t, y_{1} - 2 b t, z_{1} - 2 c t) with t = \frac{a x_{1} + b y_{1} + c z_{1} + d}{a^{2} + b^{2} + c^{2}}

When to use. Use to find the mirror image of a point across a plane.

Trap. Double the displacement from the point to the foot; do not use single t.

Coplanarity of Two Lines

Lines: \vec{r} = {\vec{a}}_{1} + λ {\vec{b}}_{1}, \vec{r} = {\vec{a}}_{2} + μ {\vec{b}}_{2} Coplanar if ({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2}) = 0

When to use. Use to check if two given lines lie in the same plane.

Trap. If the lines are parallel, they are always coplanar (but the triple product may be zero trivially).

Read the full chapter: Three-Dimensional Geometry for JEE, with what JEE asks and the common traps.