Vectors in Three Dimensions
Extend position, addition and scalar multiples to space, with distance and the standard unit vectors. Everything later in the spine is written in this notation.
What a learner can do afterwards
- Write a vector in component form and find its magnitude in space
- Find the distance between two points and the midpoint of the segment
- Express a vector as a combination of i, j and k
1 · Read
A drone in flight, or a character in a 3D game, needs three numbers to say where it is. A point in space needs (x, y, z), and a vector in space has three components <a, b, c>, added, subtracted, and scaled one component at a time. The magnitude is the square root of the sum of the squared components: <1, 2, 2> has length 3, and <3, 4, 12> has length 13.
The distance between two points is the magnitude of their difference. For A(1, minus 2, 3) and B(4, 2, 3) the difference is <3, 4, 0>, so the distance is 5. The midpoint averages each coordinate pair: the y midpoint of minus 2 and 2 is 0, and the x midpoint of 1 and 4 is 2.5.
The helpers i, j, k are single steps along x, y, z, so <-2, 5, minus 1> is the same as minus 2 i + 5 j - k. Dividing a non-zero vector by its magnitude gives the unit vector in the same direction: <6, 8, 0> has length 10, so its unit vector is <0.6, 0.8, 0>. Flipping every sign points the same length the opposite way.
For distance, subtract first, then square, add, and root. For midpoints, average each pair. Always check a unit vector: its squared components must sum to 1.
Three components per vector, distance from differences, midpoints from averages, and i, j, k as shorthand.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.