NCERT Mathematics, Class 12, textbook pages 377–393 (Part II)

How to help your child with Class 12 Maths Chapter 11, Three Dimensional Geometry, at home

Lines in space and angles between them, explained for parents: a 60-minute study session, a quick challenge, a 5-day plan and the mistakes to watch for.

skew lines: neither parallel nor meeting

NCERT Class 12 Maths Chapter 11 Three Dimensional Geometry: what it teaches

Your teen uses direction cosines and ratios, writes equations of lines in space in vector and Cartesian forms, finds angles between lines and the shortest distance between skew lines.

Why it matters later. 3D geometry is used in engineering, graphics and physics.

See it: this chapter in pictures

2i + 3j + 6k, |v| = 7l, m, n = 2/7, 3/7, 6/7l² + m² + n² = 1
Divide by the magnitude.
through (1, 2, 3), along 2i + j − k(x−1)/2 = (y−2)/1 = (z−3)/−1Cartesian form
Point plus direction.
skew linesshortest distance|(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|
Cross product gives the common perpendicular.

You need

  • a room with corners as axes
  • sticks for lines
  • a calculator

Tonight’s 60 minutes

  1. Find direction cosines of the line along 2i + 3j + 6k (2/7, 3/7, 6/7).
  2. Write the line through (1, 2, 3) parallel to 2i + j − k.
  3. Find the angle between two lines from their direction ratios.
  4. Find the shortest distance between two skew lines.

Words to use

  • direction cosines
  • direction ratios
  • vector equation
  • Cartesian equation
  • skew lines
  • shortest distance

Watch for

  • Using direction ratios as cosines. Divide by the magnitude to get cosines.
  • Assuming non-parallel lines intersect. In 3D they may be skew.

What it sounds like at home

A short example of the conversation. Your words will differ; the questions are what matter.

YouDirection cosines of 2i + 3j + 6k?

Your childMagnitude 7: 2/7, 3/7, 6/7.

A game for this week: Skew sticks

  1. Hold two sticks in space not meeting.
  2. Discuss why they don’t intersect.
  3. Measure the shortest gap.
  4. Relate to the formula.

A 5-day plan, 60 minutes a day

  1. Day 1Explain-back.
  2. Day 2Worked examples.
  3. Day 3Practice set.
  4. Day 4Real-life problem.
  5. Day 5Mixed quiz.

Everyday moments to practise

  • Aircraft flight paths.
  • 3D modelling.
  • Structural engineering.
  • Drones.

Quick check

Ask these at the end of the week. Tap to see the answer.

Direction cosines of 2i + 3j + 6k?

2/7, 3/7, 6/7.

Sum of squares of direction cosines?

1.

Can two non-parallel lines in space not meet?

Yes, skew lines.

Ready for the next chapter when your child…

  • Finds direction cosines.
  • Writes line equations in 3D.
  • Finds angles and shortest distances.

Activities, wording and illustrations are Mathify Kids’ own, following the topics of NCERT’s Mathematics (Class 12). See the full Class 12 parent guide for every chapter.

Questions parents ask

What are skew lines?

Lines in space that are neither parallel nor intersecting.

What is the sum l² + m² + n²?

1 for direction cosines.

Are planes included?

Check the current syllabus; it may focus on lines.