📚 Key Concepts
Direction Cosines & Direction Ratios- Relationship with vector components
- For a line through two points
- Angles with coordinate axes
Lines in 3D- Vector equation: r⃗ = a⃗ + λb⃗
- Cartesian equation: (x-x₁)/l = (y-y₁)/m = (z-z₁)/n
- Parametric equations: x = x₁ + at, y = y₁ + bt, z = z₁ + ct
🎯 Key Formulas
Angle between lines:- cos θ = |l₁l₂ + m₁m₂ + n₁n₂|/√[(l₁² + m₁² + n₁²)(l₂² + m₂² + n₂²)]
Direction Cosines:- cos²α + cos²β + cos²γ = 1
- Direction ratios proportional to direction cosines
⚠️ Common Mistakes to Avoid
Wrong direction ratio calculationsIncorrect angle formula applicationConfusion between skew and intersecting linesSign errors in shortest distanceMixing up vector and cartesian equationsNot checking for parallel lines
📖 Knowledge Prerequisites
Vector algebraCoordinate geometry3D visualizationBasic calculusVector operations
💡 Tips for Students
Draw 3D diagrams when possiblePractice converting between different formsRemember standard formulasCheck for special casesVerify solutions geometricallyUse systematic approach for distance problems
👉 Practice Recommedations
Finding direction cosines/ratiosWriting line equationsAngle between linesShortest distance problemsPosition vector applicationsLine intersection problemsReal-world applications
Practice each question with a timer and get instant feedback
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