Vectors in space: length, distance and direction cosines
A vector in space is the diagonal of a cuboid. Find the length of a vector, the distance between two points, a midpoint and the direction cosines from their coordinates.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- UK: Year 12
- France: grade 8
- Germany: grade 12
- Iran: grade 12
- Afghanistan: grade 11
- Syria: grade 12
- Palestine: grade 11
- Ukraine: grade 10
- Russia: grade 11
Guide
This studio follows Afghan Year 11 chapter 7 (vectors in space), Iran's Year 12 geometry chapter 3 and A level vectors. In 3D coordinates every point has three coordinates and a vector is written a = xi + yj + zk. For example 236 is 2i + 3j + 6k.
The length of a vector is the diagonal of a cuboid whose edges are its components: |a| = √(x2 + y2 + z2). The distance between points A and B is the length of AB = OB − OA, and each coordinate of the midpoint is the mean of the matching coordinates.
If α, β and γ are the angles the vector makes with the x, y and z axes, each cosine is that axis's component divided by the length. The squares of the three cosines add up to 1. Common mistakes: adding the components instead of their squares, forgetting the square root, and subtracting coordinates in the wrong order.