Describing lines and planes in space with vectors
Three-dimensional geometry locates a point by three coordinates (x, y, z) measured along mutually perpendicular axes. A line's direction is given by its direction cosines l, m and n, the cosines of the angles it makes with the three axes, which always satisfy l² + m² + n² = 1. Any three numbers proportional to them are called direction ratios. The line through the point with position vector a and parallel to the vector b is r = a + λb; in Cartesian form it is (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
The angle between two lines follows from the dot product of their direction vectors. Two lines in space may intersect, be parallel, or be skew, meaning they neither meet nor are parallel. The shortest distance between skew lines r = a₁ + λb₁ and r = a₂ + μb₂ is |(a₂ − a₁) · (b₁ × b₂)| ÷ |b₁ × b₂|. A plane can be written as r · n = d, where n is a normal vector, or as ax + by + cz = d, and the distance of the point (x₁, y₁, z₁) from the plane ax + by + cz + d = 0 is |ax₁ + by₁ + cz₁ + d| ÷ √(a² + b² + c²).
Why it matters
Three-dimensional geometry is the language of computer graphics, robotics, aircraft navigation and structural engineering, where positions and directions in space must be calculated precisely. It brings vectors and algebra together, and it is a regular source of Class 12 board and JEE questions on angles, distances and intersections of lines and planes.
Key terms
- Direction cosines
- The cosines of the angles a line makes with the x-, y- and z-axes.
- Direction ratios
- Any three numbers proportional to a line's direction cosines.
- Skew lines
- Lines in space that are neither parallel nor intersecting.
- Normal vector
- A vector perpendicular to every line lying in a given plane.