Ch 11 covers 3D geometry using vectors: direction cosines and ratios, equations of lines and planes in 3D space, angles between them, and the shortest distance between skew lines.
Direction cosines (l,m,n) of a line satisfy l²+m²+n² = 1. Direction ratios (a,b,c) are proportional to direction cosines. Angle between two lines: cos θ = |l₁l₂+m₁m₂+n₁n₂|.
Vector form: r⃗ = a⃗ + λb⃗ (passing through point a⃗ in direction b⃗). Cartesian form: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c.
Vector form: r⃗·n⃗ = d (normal n⃗). Cartesian: ax+by+cz+d = 0. Distance from point (x₁,y₁,z₁) to plane: |ax₁+by₁+cz₁+d|/√(a²+b²+c²).
Download: https://ncert.nic.in/textbook/pdf/lemh205.pdf | Complete book Part II: https://ncert.nic.in/textbook/pdf/lemh2ps.zip
Skew lines are lines in 3D space that are neither parallel nor intersecting — they lie in different planes. The shortest distance between skew lines can be found using the cross product of their direction vectors.
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