Review: vectors in space, lines and planes
Weeks 8 to 14 start with points and vectors and end with lines and planes, and the same few ideas return. Vectors: the norm, the unit vector A/‖A‖ and the direction cosines. The scalar product A · B gives angles and the test for perpendicular vectors; the vector product A × B gives a vector perpendicular to both, areas, and parallel vectors when it is zero; the scalar triple product gives volumes and coplanar vectors. A line is a point and a direction vector, and a plane is a point and a normal vector. For two lines use the angle formula with their directions, then equate coordinates with different parameters to decide between meeting and skew. For a plane, find a normal: given directly, from AB × AC, or from d₁ × d₂. Distances use a perpendicular: point to line through the foot of the perpendicular or ‖AB × d‖/‖d‖, and point to plane through |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²). A line meets a plane after putting its general point into the equation; two planes meet in a line with direction n₁ × n₂. Always check the answer in the original data.
| task | tool |
|---|---|
| norm, unit vector, direction cosines | ‖A‖ = √(x² + y² + z²); A/‖A‖ = (ℓ, m, n); ℓ² + m² + n² = 1 |
| angle between vectors; perpendicular | cos θ = A · B/(‖A‖‖B‖); A · B = 0 |
| a vector perpendicular to A and B; areas | A × B (3 × 3 determinant); triangle area = ½‖A × B‖ |
| volume; coplanar vectors | |A · (B × C)|; A · (B × C) = 0 |
| a line | r = A + t d; x = x₁ + at …; (x − x₁)/a = (y − y₁)/b = (z − z₁)/c |
| two lines | cos θ = |d₁ · d₂|/(‖d₁‖‖d₂‖); equate coordinates with s and t: meet or skew |
| a plane | n · r = n · A; ax + by + cz + d = 0; n = AB × AC, or d₁ × d₂ |
| distances | point–line: ‖AB × d‖/‖d‖; point–plane: |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²) |
| planes and lines meeting | line–plane: (n · d)t = k − n · A; planes: d = n₁ × n₂; angle between planes from n₁ · n₂ |
Plan a question trail. Pick the data ‘three points A, B, C’. List five different questions you can ask about them, each with its tool, from this unit (angle, area, plane, distance from the origin to the plane, a line through two points). Solve one with numbers of your choice.