---
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: 2026-09-12
---

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# 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

## Lesson: Adding vectors and stretching them

A column vector says how far right and how far up to go. To add, combine tops with tops and bottoms with bottoms. So (2, 3) plus (1, 4) is (3, 7). Draw the sum on a grid to check the arrow chain.

**Example.** Routes add tip to tail. Walking vector a from A to B, then vector b from B to C, lands you at a plus b from A to C. Subtraction flips one arrow: a minus b means a plus a reversed b.

A scalar stretches or shrinks an arrow. The 2 in 2a doubles the length, and a minus sign spins it round. So minus 2a doubles the length of a and reverses its direction: a vector of length 5 becomes length 10.

**Tip.** Ask whether direction matters to spot a vector. Displacement and force point somewhere, so routes need vector addition. Plain arithmetic fits size-only quantities like mass.

**Recap.** Add columns top with top, chain routes tip to tail, and let scalars stretch or flip arrows.

## 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-)