3D Geometry - How It Appears in JEE

8 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Find Direction Cosines / Direction Ratios

Pattern

Find the DCs or DRs of a line given two points, or convert between DCs and DRs

How to recognize

Asks for direction cosines, direction ratios, or the angles a line makes with the coordinate axes.

→ DRs from two points: (x2-x1, y2-y1, z2-z1). To get DCs, divide each DR by sqrt(a^2 + b^2 + c^2). Verify l^2 + m^2 + n^2 = 1.

Equation of a Line Through Points

Pattern

Write the equation of a line passing through given point(s) with given direction or two points

How to recognize

Asks for the equation of a line in symmetric form, vector form, or parametric form in 3D.

→ Use (x-x1)/a = (y-y1)/b = (z-z1)/c. Find DRs from two points or the given direction. In vector form: r = a + t*b.

Equation of a Plane

Pattern

Find the equation of a plane given a point and normal, three points, or other conditions

How to recognize

Asks for the equation of a plane passing through points, containing lines, or with a given normal direction.

→ Normal form: a(x-x1) + b(y-y1) + c(z-z1) = 0. Three-point form: use the cross product of two in-plane vectors to find the normal. Intercept form: x/a + y/b + z/c = 1.

Angle Between Lines or Planes

Pattern

Find the angle between two lines, two planes, or a line and a plane

How to recognize

Mentions angle between two lines (using DRs), angle between two planes (using normals), or angle between a line and a plane.

→ Two lines: cos(theta) = |a1a2+b1b2+c1c2| / (product of magnitudes). Two planes: same formula using normals. Line and plane: use sin(theta) instead of cos(theta).

Distance from Point to Plane

Pattern

Find the perpendicular distance from a point to a given plane

How to recognize

Asks for the distance or perpendicular distance from a point to a plane, or distance between parallel planes.

→ Use d = |ax1 + by1 + cz1 + d| / sqrt(a^2 + b^2 + c^2). For distance between parallel planes, pick any point on one and find its distance to the other.

Shortest Distance Between Lines

Pattern

Find the shortest distance between two skew lines or check if lines intersect

How to recognize

Mentions shortest distance, skew lines, or asks whether two lines in 3D intersect.

→ Use d = |(a2-a1).(b1 x b2)| / |b1 x b2|. If d = 0, lines are coplanar (intersecting or parallel). For parallel lines, use the cross product formula for distance.

Foot of Perpendicular and Image

Pattern

Find the foot of perpendicular or image of a point in a line or plane

How to recognize

Asks for foot of perpendicular from a point to a line or plane, or the reflection/image of a point.

→ Write the line through the point parallel to the normal. Find the parameter by substituting into the plane equation. For image, double the parameter value. For foot on a line, use the dot product condition.

Intersection Line of Two Planes

Pattern

Find the line of intersection of two planes or a plane through the intersection

How to recognize

Asks for the equation of the line where two planes meet, or a plane passing through the intersection of two given planes.

→ For the intersection line: the direction is n1 x n2. Find a point by setting one coordinate to zero and solving the two plane equations. For a family of planes: use P1 + lambda*P2 = 0.