Direction cosines condition, l² + m² + n² = 1, Direction cosines from direction ratios, l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²), Direction ratios from two points, x₂−x₁, y₂−y₁, z₂−z₁, Distance between two points, PQ = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²], Direction cosines through two points, ((x₂−x₁)/PQ, (y₂−y₁)/PQ, (z₂−z₁)/PQ), Vector equation of a line, r = a + λb, Vector equation of a Line through two points, r = a + λ(b−a), Cartesian form of a line, (x−x₁)/a = (y−y₁)/b = (z−z₁)/c, Line using direction cosines, (x−x₁)/l = (y−y₁)/m = (z−z₁)/n, Angle between two lines using direction ratios, cosθ = (a₁a₂+b₁b₂+c₁c₂)/(√(a₁²+b₁²+c₁²)√(a₂²+b₂²+c₂²)), Angle between two lines using direction cosines, cosθ = |l₁l₂ + m₁m₂ + n₁n₂|, Perpendicular lines condition, a₁a₂ + b₁b₂ + c₁c₂ = 0, Parallel lines condition, a₁/a₂ = b₁/b₂ = c₁/c₂, Skew lines, Lines neither parallel nor intersecting, Shortest distance between skew lines, d = |((b₁×b₂)·(a₂−a₁))| / |b₁×b₂|, Shortest distance between parallel lines, d = |b×(a₂−a₁)| / |b|, Vector angle formula, cosθ = (b₁·b₂)/(|b₁||b₂|), Coordinate axes direction cosines, x-axis (1,0,0), y-axis (0,1,0), z-axis (0,0,1), Direction cosines with angles, cos alpha, cos beta, cos Gama

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