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CBSE Class 12 · Mathematics

Three Dimensional Geometry

Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.

This lesson follows the official chapter “Three Dimensional Geometry” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 12
  • Medium level
  • 8 concepts

1Direction Cosines and Direction Ratios of a Line

Direction cosines are cosines of the angles a line makes with the axes as taught; direction ratios are any triple proportional to them. 2,3,6 as ratios: the cosines are 2/7, 3/7, 6/7 because 4+9+36=49. Cosines square-sum to 1.

Using 2,3,6 as cosines without dividing by 7 is a miss.

Figure. Direction cosines are the cosines of the angles a line makes with the three axes. Direction ratios are any triple in the same proportion.

How it works

  1. Write the ratios l:m:nThe given.
  2. Divide by the length of that tripleThe cosines.
  3. Check squares sum to 1The test.

Cosines from 2,3,6

Direction ratios 2,3,6. Find the cosines.

  • Lengthsqrt(4+9+36)=7
  • Cosines2/7, 3/7, 6/7
  • Check4/49+9/49+36/49=1

Pro tip. Ratios over their length.

2Equation of a Line in Space

Equation of a line in space: through a point with direction ratios, or through two points, as taught. Vector form r = a + λb. A line needs a point and a direction — not a single coordinate.

Writing z=0 as “the line” with no direction is a miss.

Figure. A line in space needs a point and a direction. The parametric form is the point plus a scalar times the direction vector. The dashed parts are the same line, both ways.

How it works

  1. Name a point and a direction (or two points)The data.
  2. Write the taught form (vector or Cartesian)The equation.
  3. Keep λ as the parameterThe run.

3Angle between Two Lines

Angle between two lines: cosθ from the direction-cosine (or ratio) formula as taught. Parallel means proportional ratios; perpendicular means the dot of ratios is 0. Angle is a number, not a drawing.

Calling any two space-lines perpendicular because they “look 3-d” is a miss.

Figure. The angle between two lines is the angle between their direction vectors. Cos θ is the absolute value of the normalised dot product, so the acute angle is the one the exam usually wants.

How it works

  1. Copy the two ratio-triplesThe given.
  2. Form the taught cosθ (or test parallel/perp)The angle.
  3. Keep a 0-dot as perpendicularA test.

4Shortest Distance between Two Lines

Shortest distance between two lines: the taught formula using the join of points and the two directions (skew vs intersecting). SD is a length; 0 means they meet or are parallel-coplanar as framed. One skew-case mention is enough.

Using the 2-d point-to-line formula unchanged is a steal.

Figure. The shortest distance between two skew lines is the length of their common perpendicular. If the lines meet, that length is zero. If they are parallel, any joining perpendicular has the same length.

How it works

  1. Name the two lines (point+direction each)The given.
  2. Apply the taught SD writeThe distance.
  3. Read 0 as they meet (or the parallel case as taught)Honest.

5A definition is a test you can run

3-d line-word is a test: cosines, a line-equation, an angle, or SD. If you only say “space”, you have a heading.

A cube-sticker is not the test.

Figure. l, m, n are direction cosines only when their squares add to 1. A direction-ratio triple <a,b,c> becomes cosines after dividing by its length.

How it works

  1. Name ratios, the equation, angle, or SDThe object.
  2. Give the school sentenceThe test.
  3. Then the word has contentThe definition ran.

6Name the given before the unknown

The given is a point and ratios, or two lines. The unknown is the equation or the angle. Copy 2,3,6 before you use 2,3,5.

Using 2+3+6=1 as the cosine-check is a silent skip of squares.

Figure. Write the given point and the given direction ratios before you write the symmetric equations. The unknown is the line, not those two facts.

How it works

  1. Copy the point and the tripleThe given.
  2. Name equation, angle, or SDThe unknown.
  3. Then computeGiven first.

7One worked case is enough at this class

One worked case is enough: cosines 2/7, 3/7, 6/7. Do not stack four skew-distances. The case teaches ratios-to-cosines.

A poster of ten axes is not more science.

Figure. Worked case: line through A(1,2,3) with direction ratios <2,3,6>. The cosines are 2/7, 3/7, 6/7 because 4+9+36=49. One numbered line is enough to hold the method.

How it works

  1. Take one tripleThe case.
  2. Divide by its lengthThe teach.
  3. Stop — one caseThis class.

8A miss: swapping the school name for the picture

A miss: swapping the school name for a picture. “Direction cosine” is a number-triple — a glowing axis is a setting.

A pretty 3-d box without l,m,n is a mute picture.

Figure. A plane slope m is two numbers. A space line needs three direction ratios. Swapping the 2-D picture for the 3-D name is the miss.

How it works

  1. Keep cosine/ratio as the nameThe science.
  2. Use a box only as a settingHonest.
  3. Refuse a swap of name for glowThe miss named.
Direction ratios 2,3,6 give cosines
  1. 2/7, 3/7, 6/7
  2. 2,3,6
  3. 2/11, 3/11, 6/11

Divide by 7.

Notes

  • The official chapter title is “Three Dimensional Geometry”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • cosines = ratios / length of the triple
  • cosines square-sum to 1

Recap

Hold these pegs from the official chapter “Three Dimensional Geometry”. The wording is ExamMaster’s teaching, not a textbook recap.

Direction Cosines and Direction Ratios of a Line
Direction cosines are cosines of the angles a line makes with the axes as taught; direction ratios are any triple proportional to them.
Equation of a Line in Space
Equation of a line in space: through a point with direction ratios, or through two points, as taught.
Angle between Two Lines
Angle between two lines: cosθ from the direction-cosine (or ratio) formula as taught.
Shortest Distance between Two Lines
Shortest distance between two lines: the taught formula using the join of points and the two directions (skew vs intersecting).
A definition is a test you can run
3-d line-word is a test: cosines, a line-equation, an angle, or SD.
Name the given before the unknown
The given is a point and ratios, or two lines.

Practise Three Dimensional Geometry

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