--- title: "Proving Geometry Facts with Vectors" description: "Show that one vector expression is a multiple of another, and use that to prove lines are parallel or that three points lie on a straight line." canonical: https://lightmysky.com/learn/mathematics/proving-geometry-facts-with-vectors-mt_A00MA7Ecgf source: https://lightmysky.com/learn/mathematics/proving-geometry-facts-with-vectors-mt_A00MA7Ecgf.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`. # Proving Geometry Facts with Vectors Show that one vector expression is a multiple of another, and use that to prove lines are parallel or that three points lie on a straight line. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/proving-geometry-facts-with-vectors-mt_A00MA7Ecgf ## Ready when they can - Write a route in terms of the two given vectors before comparing anything - Argue that one vector is a multiple of another, so the lines are parallel - Use a shared point plus parallel vectors to show three points are in a line ## Lesson: Proving lines parallel with vectors Write every route with the vectors you know. From A to B, undo the trip to A, then do the trip to B. With OA as a and OB as b, vector AB is b minus a. The midpoint M of AB sits at half of a plus b from O. **Example.** One vector that is a plain number times another points the same way. PQ of 2a plus 6b is exactly 2 times RS of a plus 3b, so the lines are parallel. Match the a parts and the b parts to find the number: 6a minus 3b is 1.5 times 4a minus 2b. Parallel alone is not enough for one straight line. Add a shared point: equal vectors through Q lock P, Q and R onto a single line. State the shared point in your proof, or the argument falls short. **Example.** The midpoint connector proof ties it together. In a triangle, the route joining two midpoints works out to half of the third side vector. Half the length and the same direction means parallel, which is exactly what was claimed. **Recap.** Write routes with known vectors, test for a number multiple, and add a shared point for collinearity. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Constructing mathematical arguments](https://lightmysky.com/learn/mathematics/constructing-mathematical-arguments-mt_j5YqQnN6xe) - [Vectors: Notation, Addition and Scalar Multiples](https://lightmysky.com/learn/mathematics/vectors-notation-addition-and-scalar-multiples-mt_NkqmrL7pLe) ## Opens up - [Equilibrium and Newton's First Law](https://lightmysky.com/learn/mathematics/equilibrium-and-newtons-first-law-mt_e6RFRQkRPo) - [Proof by Deduction and Exhaustion](https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f)