Vector Algebra and Three-Dimensional Geometry
What is tested
This domain covers dot and cross products, projections, direction ratios and cosines, equations of lines and planes, angles between them, shortest distances, and coplanarity of vectors.
Core rules
- a·b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃; the projection of a on b is (a·b)/|b|.
- a×b is perpendicular to both a and b, with |a×b| = |a||b|sin θ, and b×a = −(a×b).
- A unit vector along a is a/|a|; direction cosines satisfy l² + m² + n² = 1.
- Line through a with direction b: r = a + λb. Plane with normal n through a: (r − a)·n = 0.
- Angle between a line and a plane: sin θ = |b·n|/(|b||n|); between two planes use their normals.
- Shortest distance between skew lines = |(a₂ − a₁)·(b₁×b₂)|/|b₁×b₂|.
Worked example
a = 2i − j + k, b = i + j + 2k. Then a·b = 2(1) + (−1)(1) + 1(2) = 3, while |a| = √(4+1+1) = √6 and |b| = √(1+1+4) = √6. So cos θ = 3/6 = 1/2 and θ = π/3. The projection of a on b is 3/√6 = √6/2.
Marks-losing traps
Writing b×a in place of a×b; forgetting to normalise when a unit vector is demanded; using cosine instead of sine for a line–plane angle; confusing direction ratios with direction cosines; omitting the modulus in the shortest-distance formula.