--- title: "Vectors in Three Dimensions" description: "Extend position, addition and scalar multiples to space, with distance and the standard unit vectors. Later topics in this sequence use 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: 2026-11-12 --- > **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. Later topics in this sequence use 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 - Normalize a non-zero vector by dividing each component by its magnitude - Subtract the coordinates of two points to form the vector between them ## Lesson: Finding your way in three dimensions In the plane a vector had two components, (a, b). In space it has three, written . Find its magnitude by squaring each component, adding, and taking the square root. **Example.** For example, <4, 4, 7> has magnitude, or length, equal to the square root of 16 + 16 + 49, which is 9. The distance between two points is the magnitude of their difference. For A(-2, 1, 4) and B(1, 5, 4), the difference is <3, 4, 0>, so the distance is 5. **Example.** A midpoint averages matching coordinates. For R(-3, 2, 6) and S(5, 8, 0), the y-coordinate is (2 + 8) divided by 2, which is 5. To find a unit vector in the same direction, divide every component by the magnitude. The vector <5, 12, 0> has magnitude 13, so its unit vector is <5/13, 12/13, 0>. To point the other way, first normalize the vector, then change every sign. The vector <1, 2, 2> has magnitude 3, so its opposite unit vector is <-1/3, -2/3, -2/3>. The standard unit vectors i, j and k are axis helpers pointing 1 unit along x, y and z in turn. Each component multiplies its helper, so <4, -3, 2> is 4i - 3j + 2k. **Recap.** Three components per vector, distance from differences, midpoints from averages, and i, j, k as shorthand. *The lesson and quiz items were written for this dataset by its content-upgrade pipeline. The lesson examples, the unit vector section and quiz item q2 were later revised.* ## 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)