Playbook
3D Geometry
Lines and planes in space: the foot, image and distance, and the shortest distance between skew lines. Each task is one formula once the direction ratios are read correctly.
- Questions in the bank
- 268
- q/paper in 2025–26
- 1.76
- Numeric answer
- 28%
- Notes pages
- 6
Tier: Cornerstone
When you’ll see it
A line in symmetric or vector form, a plane's equation, or a point to be dropped onto, reflected in or measured from either.
How this chapter is tested
Two objects run through the whole chapter: the general point of a line, (x₁ + at, y₁ + bt, z₁ + ct), and the normal of a plane. Almost every question writes one of them and imposes a condition with the other.
The shortest distance between skew lines and the foot or image of a point in a line are the most common single tasks, and each is one formula once the directions are read correctly. Plane questions are mostly about finding the normal: a cross product of two directions, or the λ in P₁ + λP₂ = 0.
Time goes in reading the equations. A line written with (2 − x)/3 has direction ratio −3 for x, and two parallel planes must share the same a, b, c before their constants are compared. The chapter is vector algebra in coordinates: the dot, cross and triple products carry it.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Direction ratios and lines
Divide ratios by their length to get direction cosines; the cross product of two directions gives a line perpendicular to both.
Foot, image and distance from a line
Make the join from the point to the line's general point perpendicular to the direction; the image is 2M − P.
Skew and parallel lines
Shortest distance |(a₂ − a₁) · (d₁ × d₂)| / |d₁ × d₂|; for non-parallel lines, zero means they meet.
The equation of a plane
Point and normal, three points, intercepts, or the family P₁ + λP₂ = 0 through a line of intersection.
Distance, foot and image in a plane
|ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²); the foot and image lie along the normal, found with one parameter.
Lines meeting planes
Put the line's general point into the plane; the angle uses sin θ = |d · n| / (|d||n|).
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.
Denominators read as direction ratios
In (2 − x)/3 = (3y − 2)/k the ratio for x is −3 and for y is k/3. Rewrite so x, y and z each carry coefficient +1 first.
Planes not matched before subtracting
2x + y − 2z = 1 and 4x + 2y − 4z = 11 are parallel, but their distance is |11/2 − 1| / 3 = 3/2, not |11 − 1| / 3.
Cosine for the line–plane angle
The dot product with the normal gives the angle with the normal. The angle with the plane is its complement, so the formula has sin θ.
Two equations are not enough
Any two coordinates can be solved for the two parameters. The lines meet only if the third coordinate also agrees, and each line needs its own parameter.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.
3D Geometry notesDrill every 3D Geometry question
268 questions from the bank, across 6 subtopics.
Drill one subtopic at a time
The 6 subtopics, in teaching order.
- Direction Cosines and Equations of LinesDrill Direction Cosines and Equations of Lines
- Foot, Image and Distance from a LineDrill Foot, Image and Distance from a Line
- Shortest Distance, Intersection and Coplanar LinesDrill Shortest Distance, Intersection and Coplanar Lines
- Equation of a PlaneDrill Equation of a Plane
- Distance, Foot and Image in a PlaneDrill Distance, Foot and Image in a Plane
- Lines Meeting PlanesDrill Lines Meeting Planes
Related playbooks
Often paired with this one — the technique or the trap overlaps. Drill these next.