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CBSE · Class 12 · Mathematics

Direction Cosines and Direction Ratios of a Line (Ch 11: 11.2)

2 concepts · 16 practice questions with answers · Free to read

Direction cosines and direction ratios

From daily life

A signboard says "head north-east". The directions of a line in space are described by the cosines of the angles it makes with the three axes, or by any three numbers in the same proportion.

Simple definition

If a line makes angles α, β, γ with the axes, then l = cos α, m = cos β, n = cos γ are its direction cosines, and l² + m² + n² = 1. Any numbers a, b, c proportional to l, m, n are direction ratios. To get direction cosines, divide by √(a² + b² + c²). The opposite direction has all signs changed.

Formula: l = a/√(a² + b² + c²), m = b/√(a² + b² + c²), n = c/√(a² + b² + c²), l² + m² + n² = 1

Rule 1: Ratios to cosines — Divide each ratio by the square root of the sum of squares.

  • Easy: The x-axis has direction cosines 1, 0, 0.
  • Medium: Ratios 2, 3, 6 give cosines 2/7, 3/7, 6/7, since √(4 + 9 + 36) = 7.
  • Exam-type: Ratios 3, 4, 12 give cosines 3/13, 4/13, 12/13.

Rule 2: Using l² + m² + n² = 1 — Find a missing cosine.

  • Easy: Ratios 1, 1, 1 give cosines 1/√3 each.
  • Medium: If l = 1/2 and m = 1/√2, then n² = 1 − 1/4 − 1/2 = 1/4, so n = ±1/2.
  • Exam-type: The opposite direction of the line with cosines 2/7, 3/7, 6/7 has cosines −2/7, −3/7, −6/7.
Remember

Direction cosines are unique up to the sign for the opposite direction, but direction ratios can be multiplied by any non-zero number.

Common mistake

Taking direction ratios as direction cosines without checking that the squares add to 1.

Memory clue

Ratios to cosines: divide by the length.

Practice questions

Easy
  1. 1Write the direction cosines of the x-axis.
  2. 2Find the direction cosines of the line with direction ratios 2, 3, 6.
  3. 3Find the direction cosines of the line with direction ratios 1, 1, 1.
Medium
  1. 4Find the direction cosines of the line joining the origin to (3, 4, 12).
  2. 5If l = 1/2 and m = 1/√2, find n.
  3. 6Can 1/2, 1/2, 1/2 be the direction cosines of a line?
Challenging
  1. 7Write the direction cosines of the opposite direction of the line with cosines 2/7, 3/7, 6/7.
  2. 8Find the direction cosines of the line through (1, 2, 3) and (3, −1, 9).
Show answers (try them yourself first)
  1. 11, 0, 0
  2. 22/7, 3/7, 6/7
  3. 31/√3, 1/√3, 1/√3
  4. 43/13, 4/13, 12/13
  5. 5n² = 1/4, so n = ±1/2
  6. 6No, since 1/4 + 1/4 + 1/4 = 3/4, which is not 1
  7. 7−2/7, −3/7, −6/7
  8. 8Ratios 2, −3, 6, so cosines 2/7, −3/7, 6/7

Line through two points

From daily life

A cable goes from the top of one tower to the top of another. To describe its direction, subtract the coordinates of the two ends, and the three differences are the direction ratios.

Simple definition

For the line through P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), the direction ratios are x₂ − x₁, y₂ − y₁, z₂ − z₁. Divide by the distance PQ to get the direction cosines.

Formula: Direction ratios = (x₂ − x₁, y₂ − y₁, z₂ − z₁), direction cosines = ratios / PQ

Rule 1: Subtract, then divide — Differences first, then the length.

  • Easy: For (2, 3, 4) and (5, 7, 16) the ratios are 3, 4, 12, PQ = 13 and the cosines are 3/13, 4/13, 12/13.
  • Medium: The line through the origin and (2, −2, 1) has cosines 2/3, −2/3, 1/3.
  • Exam-type: The points (1, 2, 3), (2, 4, 6), (3, 6, 9) are collinear, because the ratios of AB and BC are both 1, 2, 3.

Rule 2: Special lines — Equally inclined means equal cosines.

  • Easy: A line equally inclined to all axes has cosines ±1/√3, and each angle is cos⁻¹(1/√3), about 54.7°.
  • Medium: Cosines 1/2, 1/2, 1/√2 satisfy l² + m² + n² = 1 and make 60°, 60°, 45° with the axes.
  • Exam-type: If l = 2/7 and m = 3/7, then n² = 36/49, so n = ±6/7.
Remember

Collinear points give proportional direction ratios for AB and BC.

Common mistake

Subtracting in different orders for the different coordinates.

Memory clue

End minus start, in every coordinate.

Practice questions

Easy
  1. 1Find the direction ratios and cosines of the line through (2, 3, 4) and (5, 7, 16).
  2. 2Find the direction cosines of the line joining the origin and (2, −2, 1).
  3. 3Show that (1, 2, 3), (2, 4, 6), (3, 6, 9) are collinear.
Medium
  1. 4Find the direction cosines of a line equally inclined to the three axes.
  2. 5Find the distance between (2, 3, 4) and (5, 7, 16).
  3. 6A line has direction cosines 1/2, 1/2, 1/√2. What angle does it make with the z-axis?
Challenging
  1. 7If l = 2/7 and m = 3/7, find n.
  2. 8Find the direction cosines of the line through (1, 1, 1) and (3, 3, 3).
Show answers (try them yourself first)
  1. 1Ratios 3, 4, 12 and cosines 3/13, 4/13, 12/13
  2. 22/3, −2/3, 1/3
  3. 3Both AB and BC have ratios 1, 2, 3
  4. 4±1/√3, ±1/√3, ±1/√3
  5. 5√(9 + 16 + 144) = 13
  6. 6cos γ = 1/√2, so 45°
  7. 7n² = 1 − 4/49 − 9/49 = 36/49, so n = ±6/7
  8. 8Ratios 2, 2, 2, so cosines 1/√3, 1/√3, 1/√3