Three Dimensional Geometry – Class 12 Mathematics notes, important questions and MCQs

Chapter overview

Lines in space are described by vector and Cartesian equations using direction ratios. Angles between lines and shortest distances follow from vectors.

Lines

• Direction cosines l, m, n (l² + m² + n² = 1); direction ratios a, b, c proportional to them.
• Line through a point a parallel to b: r = a + λb; Cartesian (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
• Line through two points: (x − x₁)/(x₂ − x₁) = (y − y₁)/(y₂ − y₁) = (z − z₁)/(z₂ − z₁).

Angles and distances

• Angle between lines: cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(a₁² + b₁² + c₁²) √(a₂² + b₂² + c₂²)).
• Perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0.
• Shortest distance between skew lines r = a₁ + λb₁, r = a₂ + µb₂: |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|.

Key terms

Skew lines

lines that are neither parallel nor intersecting

Direction ratios

numbers proportional to the direction cosines of a line

Coplanar lines

lines lying in the same plane; two lines are coplanar if they intersect or are parallel

Formulas

• l² + m² + n² = 1
• SD = |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|

Important questions with answers

Quick revision

• Skew lines exist only in 3D.
• Perpendicular ⇒ sum of products of DRs = 0.
• Direction cosines of the x-axis: 1, 0, 0.

Exam tips

• Parallel lines have proportional direction ratios.
• For the shortest distance between parallel lines r = a₁ + λb and r = a₂ + µb use |b × (a₂ − a₁)|/|b|.

Practice MCQs

1. Direction cosines of the y-axis are

   1. 1, 0, 0
   2. 0, 0, 1
   3. 0, 1, 0
   4. 1, 1, 0
   Show answer

   Answer: (c) 0, 1, 0. Along ĵ.

2. For direction cosines l, m, n, l² + m² + n² equals

   1. 0
   2. 1
   3. 2
   4. 3
   Show answer

   Answer: (b) 1. Identity.

3. Lines with DRs (1, 2, 3) and (2, −1, 0) are

   1. parallel
   2. perpendicular
   3. the same
   4. at 45°
   Show answer

   Answer: (b) perpendicular. 2 − 2 + 0 = 0.

4. Lines with DRs (1, 2, 2) and (2, 4, 4) are

   1. perpendicular
   2. at 60°
   3. skew always
   4. parallel
   Show answer

   Answer: (d) parallel. Proportional.

5. Angle between the lines with direction ratios (1, 1, 0) and (1, 0, 1) is

   1. 30°
   2. 45°
   3. 60°
   4. 90°
   Show answer

   Answer: (c) 60°. cos θ = 1/2.

6. Direction cosines of the line with DRs (2, −1, 2) are

   1. (2, −1, 2)
   2. (2/9, −1/9, 2/9)
   3. (1/3, 1/3, 1/3)
   4. (2/3, −1/3, 2/3)
   Show answer

   Answer: (d) (2/3, −1/3, 2/3). Divide by 3.

7. Lines that are neither parallel nor intersecting are called

   1. coplanar
   2. coincident
   3. concurrent
   4. skew
   Show answer

   Answer: (d) skew. Skew lines.

8. The line (x − 1)/2 = (y + 1)/3 = (z − 2)/4 passes through

   1. (1, −1, 2)
   2. (2, 3, 4)
   3. (−1, 1, −2)
   4. (0, 0, 0)
   Show answer

   Answer: (a) (1, −1, 2). Put λ = 0.