Class 12 Maths NCERT Solutions Chapter 11 Three Dimensional Geometry | Boundless Maths
NCERT Solutions Class 12 Maths Chapter 11 · Unit IV · Vectors & 3D Geometry

Class 12 Maths NCERT Solutions Chapter 11: Three Dimensional Geometry

Free, step-by-step Class 12 Maths NCERT Solutions Chapter 11 for all three parts of this chapter — direction cosines and direction ratios of a line, the vector and Cartesian equation of a line, angle between two lines, and shortest distance between skew or parallel lines. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

3Exercises (incl. Misc.)
25Total Questions
2026-27CBSE Syllabus
100%Solved

Class 12 Maths NCERT Solutions Chapter 11 — Overview

Three Dimensional Geometry takes the vector algebra you built in Chapter 10 and points it directly at lines in 3D space. Instead of Class 11's purely Cartesian approach, this chapter uses vectors to describe a line's direction, write its equation, measure the angle between two lines, and — the technique that feels genuinely new — find the shortest distance between two lines that never meet and aren't parallel either: skew lines.

The chapter is compact but dense: two exercises plus a Miscellaneous Exercise, 25 questions in total, and every one of them leans on the same handful of ideas. Get comfortable converting between direction cosines and direction ratios, writing a line's equation in both vector and Cartesian form, and you'll recognise almost every question type CBSE sets from this chapter — including the shortest-distance formula, which is one of the more reliably-scoring 5-mark questions on the paper once you've drilled the cross-product mechanics a few times.

How the Chapter Builds

One Idea Leads to the Next

1

Direction Cosines & Ratios

l, m, n describe a line's direction exactly; a, b, c describe it up to scale. Exercise 11.1.

→
2

Equation of a Line

A point plus a direction fixes a line completely — in both vector form r = a + λb and Cartesian form. Exercise 11.2.

→
3

Angle Between Lines

Compare direction vectors (or direction ratios) with a dot product to test for perpendicularity or find the angle. Exercise 11.2.

→
4

Shortest Distance

For skew lines, the cross product of their direction vectors gives the perpendicular direction — and the distance formula. Exercise 11.2 & Misc.

Quick Reference

Important Formulas — Chapter 11

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below have the full printable version for the whole syllabus.

Direction Cosines & Direction Ratios (§11.2)

Fundamental relation

l^2+m^2+n^2=1

True only for direction cosines — never assume this holds for direction ratios.

Direction cosines joining two points

\dfrac{x_2-x_1}{PQ},\ \dfrac{y_2-y_1}{PQ},\ \dfrac{z_2-z_1}{PQ}

where PQ is the distance between the two points.

Direction cosines from direction ratios

l=\dfrac{a}{\sqrt{a^2+b^2+c^2}},\ m=\dfrac{b}{\sqrt{a^2+b^2+c^2}},\ n=\dfrac{c}{\sqrt{a^2+b^2+c^2}}

Just the direction ratio vector, scaled down to unit length.

Equation of a Line (§11.3)

Vector form

\vec{r}=\vec{a}+\lambda\vec{b}

Through the point with position vector a, parallel to direction vector b.

Cartesian form

\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}

Through (x_1,y_1,z_1) with direction ratios a, b, c.

Line through two points

\vec{r}=\vec{a}+\lambda(\vec{b}-\vec{a})

Direction vector is simply the displacement from one point to the other.

Angle Between Two Lines (§11.4)

Using direction ratios

\cos\theta=\left|\dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|

The absolute value keeps θ as the acute angle between the lines.

Perpendicular lines

a_1a_2+b_1b_2+c_1c_2=0

The dot product of the direction ratios vanishes.

Parallel lines

\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}

Direction ratios of one line are a scalar multiple of the other's.

Shortest Distance Between Lines (§11.5)

Skew lines (vector form)

d=\left|\dfrac{(\vec{b_1}\times\vec{b_2})\cdot(\vec{a_2}-\vec{a_1})}{|\vec{b_1}\times\vec{b_2}|}\right|

Zero only when the lines actually intersect — otherwise this gives the true minimum gap.

Parallel lines

d=\left|\dfrac{\vec{b}\times(\vec{a_2}-\vec{a_1})}{|\vec{b}|}\right|

Uses the single shared direction vector b, since both lines are parallel to it.

Decision Guide

Which Formula Should I Use?

A quick way to decide, once you've read what the question actually gives you.

What the question gives youUse thisWhy
Two points on a lineDirection ratios / cosines of a joining vectorSubtract the coordinates first — that gives you the direction to work with everywhere else.
A point and a direction (or two points)Equation of a line (vector or Cartesian)A line is completely determined by exactly this information.
Two lines, and you need the angle or a perpendicularity/parallelism checkAngle-between-lines formula on their direction ratios or vectorsPerpendicular and parallel are just the two extreme cases of this same formula.
Two lines that are neither parallel nor intersectingSkew-line shortest-distance formulaThere's no intersection point to find — the cross product gives you the perpendicular gap instead.
Two lines confirmed parallelParallel-line distance formulaSimpler than the skew-line version, since both lines share one direction vector.
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all three exercises.

  • Confusing direction cosines with direction ratios — using unnormalised direction ratios where l² + m² + n² = 1 is actually required, or vice versa.
  • Sign errors in the cross product determinant when computing b₁ × b₂ — a single misplaced sign flips the whole shortest-distance answer.
  • Applying the skew-line distance formula to lines that are actually parallel — if the direction vectors are proportional, b₁ × b₂ = 0 and the formula breaks down; use the parallel-line formula instead.
  • Forgetting the absolute value in the angle-between-lines formula — cosθ should always come out non-negative, since the angle taken is the acute one.
  • Mixing up which point is a₁ and which is a₂ when substituting into the shortest-distance formula — the subtraction a₂ − a₁ must be consistent with which line's direction vector is b₁.
  • Not checking whether two lines actually intersect before assuming they're skew — if the shortest distance works out to exactly 0, the lines meet.
  • Arithmetic slips inside the 3×3 cross-product determinant — these are easy to rush and hard to spot; expand it carefully term by term.
  • Forgetting to double-check direction ratios are proportional in all three components, not just one or two, when testing whether two lines are parallel.
Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 11 — Choose an Exercise

11.1

Exercise 11.1

Direction cosines and direction ratios of a line — from given angles, from direction ratios, and proving collinearity · 5 questions

Solve Exercise 11.1 →
11.2

Exercise 11.2

Equation of a line, angle between lines, perpendicularity/parallelism, and shortest distance between skew or parallel lines · 15 questions

Solve Exercise 11.2 →
M

Miscellaneous Exercise

Mixed questions combining direction ratios, perpendicularity, and the shortest-distance formula in less template-like settings · 5 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Vector Algebra and Three Dimensional Geometry — plus every other chapter — in one printable set of Formula Cards.

Get Formula Cards →

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Common Questions

FAQs — Class 12 Maths NCERT Solutions Chapter 11

Quick answers about Chapter 11, Three Dimensional Geometry.

How many exercises are there in Chapter 11, Three Dimensional Geometry?

There are two main exercises — 11.1 and 11.2 — plus a Miscellaneous Exercise, totalling 25 questions across direction cosines/ratios, the equation of a line, angle between lines, and shortest distance between lines.

Is this chapter important for the board exam?

Yes — questions on the shortest distance between skew lines and the equation of a line through a point are near-certain across CBSE board papers, and this chapter builds directly on the vector algebra learned in Chapter 10.

What is the difference between direction cosines and direction ratios?

Direction cosines l, m, n are the cosines of the angles a line makes with the x, y and z axes, and always satisfy l² + m² + n² = 1. Direction ratios a, b, c are any numbers proportional to the direction cosines — they don't have to satisfy that equation, and a line has infinitely many valid sets of direction ratios.

How do I know if two lines are parallel, intersecting, or skew?

Compare their direction ratios: if they're proportional, the lines are parallel. If not, check whether the lines share a common point — if they do, they intersect, and if they don't (and aren't parallel either), they're skew, in which case you'd use the shortest distance formula rather than looking for an intersection point.

What should I revise before starting this chapter?

Make sure Chapter 10 (Vector Algebra) is solid first — direction cosines, the dot product and the cross product all carry straight over into this chapter, just applied specifically to lines in 3D space.

Where can I find the official NCERT textbook for this chapter?

Chapter 11, Three Dimensional Geometry, is from the NCERT Class 12 Mathematics textbook (Part I), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this site follow the questions exactly as they appear there.
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