--- title: "Vectors: Notation, Addition and Scalar Multiples" description: "Write vectors as columns or as bold letters, add and subtract them along a route, and multiply by a scalar to change length or reverse direction." canonical: https://lightmysky.com/learn/mathematics/vectors-notation-addition-and-scalar-multiples-mt_NkqmrL7pLe source: https://lightmysky.com/learn/mathematics/vectors-notation-addition-and-scalar-multiples-mt_NkqmrL7pLe.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: Notation, Addition and Scalar Multiples Write vectors as columns or as bold letters, add and subtract them along a route, and multiply by a scalar to change length or reverse direction. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/vectors-notation-addition-and-scalar-multiples-mt_NkqmrL7pLe ## Ready when they can - Add two column vectors and draw the result on a grid - Write a route around a shape as a sum of named vectors - Say what -2a does to the length and the direction of a - Subtract matching vector components to find a difference - Add three or more vectors component by component - Reverse a vector by multiplying every component by -1 - Use a scalar to multiply every component of a vector - Explain why changing the order of vector addition does not change the sum ## Lesson: Adding vectors and stretching them A column vector says how far right and how far up to go. Add matching entries: top with top and bottom with bottom. For example, (7, -2) plus (3, 5) is (10, 3). **Example.** Add another pair the same way. The top entries in (6, 2) and (4, 7) give 6 plus 4 = 10. The bottom entries give 2 plus 7 = 9, so the sum is (10, 9). Routes add tip to tail. If vector a takes you from A to B and vector b takes you from B to C, the whole route from A to C is a plus b. **Example.** Subtract matching entries, or add the reverse of the second vector. For (9, 6) minus (4, 2), subtract tops and bottoms to get (5, 4). A scalar multiplies every component. For -3a, the 3 triples the length and the minus reverses direction. If a has length 4, then -3a has length 12. **Example.** A positive scalar keeps direction and changes length by its size. If vector b has length 7, multiplying by 3 gives a vector with length 21 and the same direction. Check a vector sum by adding matching entries again. If the top entries total 6 and the bottom entries total 6, the result must have both components equal to 6. A different component shows an arithmetic error. **Recap.** Add matching components, chain routes tip to tail, subtract by reversing the second vector, and use scalars to stretch or reverse arrows. *The lesson and quiz items were written for this dataset by its content-upgrade pipeline. The lesson examples and quiz item q8 were later revised.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Coordinate Transformations](https://lightmysky.com/learn/mathematics/coordinate-transformations-mt_K0Y15w48SY) - [Similarity and enlargement](https://lightmysky.com/learn/mathematics/similarity-and-enlargement-mt_y-BuQAfw4B) ## Opens up - [Forces as Vectors: Resolving and Resultants](https://lightmysky.com/learn/mathematics/forces-as-vectors-resolving-and-resultants-mt_-qDRgd2g8v) - [Proving Geometry Facts with Vectors](https://lightmysky.com/learn/mathematics/proving-geometry-facts-with-vectors-mt_A00MA7Ecgf) - [Data as a Matrix: Rows, Features and the Target](https://lightmysky.com/learn/computing/data-as-a-matrix-rows-features-and-the-target-mt_KEN-TU1Gpn) - [Modelling Motion in a Straight Line](https://lightmysky.com/learn/mathematics/modelling-motion-in-a-straight-line-mt_no78_6w1rS) - [Vectors in Three Dimensions](https://lightmysky.com/learn/mathematics/vectors-in-three-dimensions-mt_o97GclnNC-)