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EAPCET Mathematics · Intermediate first year · Paper IB

Direction Cosines and Direction Ratios

Direction cosines and direction ratios describe the direction of a line in space and give the angle between two lines.

Subject
Mathematics
Taught in
Intermediate first year · Paper IB
Mathematics in the paper
80 of 160 questions

Key formulas and results

  • l2+m2+n2=1l^{2} + m^{2} + n^{2} = 1
  • A line through two points has direction ratios (x2−x1,y2−y1,z2−z1)(x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})
  • cos⁡θ=l1l2+m1m2+n1n2\cos\theta = l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}
  • Perpendicular lines: a1a2+b1b2+c1c2=0a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0; parallel lines have proportional direction ratios
  • If a line makes angles α,β,γ\alpha, \beta, \gamma with the axes, cos⁡2α+cos⁡2β+cos⁡2γ=1\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1
  • Line through (x1,y1,z1)(x_{1}, y_{1}, z_{1}) with direction ratios (a,b,c)(a, b, c): x−x1a=y−y1b=z−z1c\dfrac{x - x_{1}}{a} = \dfrac{y - y_{1}}{b} = \dfrac{z - z_{1}}{c}
  • Line through (x1,y1,z1)(x_{1}, y_{1}, z_{1}) and (x2,y2,z2)(x_{2}, y_{2}, z_{2}): x−x1x2−x1=y−y1y2−y1=z−z1z2−z1\dfrac{x - x_{1}}{x_{2} - x_{1}} = \dfrac{y - y_{1}}{y_{2} - y_{1}} = \dfrac{z - z_{1}}{z_{2} - z_{1}}

What this chapter covers

  • Direction cosines and their relation
  • Direction ratios
  • Angle between two lines
  • Parallel and perpendicular lines

How it is usually asked

  • Find the angle between two lines
  • Find direction cosines from given conditions
  • Test perpendicularity or parallelism

Understanding Direction Cosines and Direction Ratios

If a line makes angles α, β and γ with the positive x-, y- and z-axes, then l = cos α, m = cos β and n = cos γ are its direction cosines. They are the components of a unit vector along the line, so they always satisfy l² + m² + n² = 1. Reversing the direction of the line changes the sign of all three.

Direction ratios are any three numbers a, b, c proportional to l, m, n, and they are often easier to find: a line through two points has direction ratios (x₂ − x₁, y₂ − y₁, z₂ − z₁). To recover direction cosines, divide each ratio by √(a² + b² + c²) and choose a common sign. Many questions can be answered without this step.

The angle between two lines comes from the dot product: cos θ = l₁l₂ + m₁m₂ + n₁n₂, or the same expression with direction ratios divided by both magnitudes. Lines are perpendicular when a₁a₂ + b₁b₂ + c₁c₂ = 0 and parallel when the ratios are proportional. EAPCET questions test these relations and the identity cos²α + cos²β + cos²γ = 1.

Worked example

EAPCET-style question

A line makes angles of 60∘60^\circ and 45∘45^\circ with the positive xx- and yy-axes. The acute angle it makes with the positive zz-axis is

  1. (1)30°
  2. (2)45°
  3. (3)60°
  4. (4)90°

Solution

  1. Use cos⁡2α+cos⁡2β+cos⁡2γ=1\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1.
  2. cos⁡260∘=14\cos^{2}60^\circ = \tfrac{1}{4} and cos⁡245∘=12\cos^{2}45^\circ = \tfrac{1}{2}, so cos⁡2γ=1−14−12=14\cos^{2}\gamma = 1 - \tfrac{1}{4} - \tfrac{1}{2} = \tfrac{1}{4}.
  3. cos⁡γ=±12\cos\gamma = \pm\tfrac{1}{2}; the acute angle has cos⁡γ=12\cos\gamma = \tfrac{1}{2}.
  4. So γ=60∘\gamma = 60^\circ.

Answer: 60°

Shortcuts and results to remember

  • Set the symmetric form equal to λ\lambda to get the general point (x1+aλ,y1+bλ,z1+cλ)(x_{1} + a\lambda, y_{1} + b\lambda, z_{1} + c\lambda); for two lines, equate general points and solve for the two parameters to find where they meet.
  • In x−x1a=y−y1b=z−z1c\dfrac{x - x_{1}}{a} = \dfrac{y - y_{1}}{b} = \dfrac{z - z_{1}}{c}, the vector ai^+bj^+ck^a\hat{i} + b\hat{j} + c\hat{k} points along the line.

More practice questions

Practice question 1

The lines x−11=y−21=z−32\dfrac{x - 1}{1} = \dfrac{y - 2}{1} = \dfrac{z - 3}{2} and x−11/32=y−11/31=z−19/32\dfrac{x - 11/3}{2} = \dfrac{y - 11/3}{1} = \dfrac{z - 19/3}{2} meet at

  1. (1)(53,83,133)\left(\tfrac{5}{3}, \tfrac{8}{3}, \tfrac{13}{3}\right)
  2. (2)(1,2,3)(1, 2, 3)
  3. (3)(113,113,193)\left(\tfrac{11}{3}, \tfrac{11}{3}, \tfrac{19}{3}\right)
  4. (4)(2,3,5)(2, 3, 5)

Solution

  1. General points: (μ+1,μ+2,2μ+3)(\mu + 1, \mu + 2, 2\mu + 3) and (2λ+113,λ+113,2λ+193)\left(2\lambda + \tfrac{11}{3}, \lambda + \tfrac{11}{3}, 2\lambda + \tfrac{19}{3}\right)
  2. xx: 2λ+113=μ+12\lambda + \tfrac{11}{3} = \mu + 1 gives 6λ+11=3μ+36\lambda + 11 = 3\mu + 3
  3. yy: λ+113=μ+2\lambda + \tfrac{11}{3} = \mu + 2 gives 3λ+11=3μ+63\lambda + 11 = 3\mu + 6
  4. Subtracting: 3λ=−33\lambda = -3, so λ=−1\lambda = -1 and μ=23\mu = \tfrac{2}{3}; the zz-coordinates agree (133\tfrac{13}{3} each)
  5. Point =(53,83,133)= \left(\tfrac{5}{3}, \tfrac{8}{3}, \tfrac{13}{3}\right)

Answer: (53,83,133)\left(\tfrac{5}{3}, \tfrac{8}{3}, \tfrac{13}{3}\right)

Common mistakes to avoid

  • Using l + m + n = 1; the condition is on the squares, l² + m² + n² = 1.
  • Treating direction ratios as direction cosines without dividing by √(a² + b² + c²) when the question asks for actual cosines.
  • Using cos θ = a₁a₂ + b₁b₂ + c₁c₂ with direction ratios; you must divide by the product of the two magnitudes.
  • Dropping the ± when finding a direction cosine from the identity; both signs give valid lines unless the question fixes the angle as acute or obtuse.

How to prepare it

Convert direction ratios to cosines by dividing by √(a² + b² + c²) only when the question needs unit values.

Once the formulas feel familiar, test them the way the exam does: practise Direction Cosines and Direction Ratios questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise Direction Cosines and Direction Ratios chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Direction Cosines and Direction Ratios part of the EAPCET Mathematics syllabus?
Yes. Direction Cosines and Direction Ratios is taught in Intermediate first year · Paper IB Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Direction Cosines and Direction Ratios for EAPCET?
Convert direction ratios to cosines by dividing by √(a² + b² + c²) only when the question needs unit values.
How many Mathematics questions are there in EAPCET?
80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in Direction Cosines and Direction Ratios?
Using l + m + n = 1; the condition is on the squares, l² + m² + n² = 1. Treating direction ratios as direction cosines without dividing by √(a² + b² + c²) when the question asks for actual cosines. Using cos θ = a₁a₂ + b₁b₂ + c₁c₂ with direction ratios; you must divide by the product of the two magnitudes. Dropping the ± when finding a direction cosine from the identity; both signs give valid lines unless the question fixes the angle as acute or obtuse.