--- title: "Vectors in Three Dimensions" description: "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." canonical: https://lightmysky.com/learn/mathematics/vectors-in-three-dimensions-mt_o97GclnNC- source: https://lightmysky.com/learn/mathematics/vectors-in-three-dimensions-mt_o97GclnNC-.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Geometry · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/vectors-in-three-dimensions-mt_o97GclnNC- ## Ready when they can - 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 ## Lesson: Finding your way in three dimensions 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 , 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. **Example.** 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. **Tip.** 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. **Recap.** Three components per vector, distance from differences, midpoints from averages, and i, j, k as shorthand. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Pythagoras and Trigonometry in Three Dimensions](https://lightmysky.com/learn/mathematics/pythagoras-and-trigonometry-in-three-dimensions-mt_7yKxYqQHB2) - [Vectors: Notation, Addition and Scalar Multiples](https://lightmysky.com/learn/mathematics/vectors-notation-addition-and-scalar-multiples-mt_NkqmrL7pLe) ## Opens up - [Lines and Planes in Space](https://lightmysky.com/learn/mathematics/lines-and-planes-in-space-mt_kIX5ak1GRv) - [The Dot Product: Angles and Projections](https://lightmysky.com/learn/mathematics/the-dot-product-angles-and-projections-mt_yFBcyE4sFw)