Skip to main content
MathidiaMathematics Academy
Vectors and geometric proof
Lesson 46 of 75
Introduction

Vector arithmetic

Objective. Combine and scale column and geometric vectors.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Vector arithmetic covers adding, subtracting and scaling vectors, whether given as column vectors or described geometrically with magnitude and direction. To add or subtract column vectors, combine corresponding components; to scale a vector by a number k, multiply every component by k. For a = (3, 2) and b = (−1, 4), a + 2b = (3 + (−2), 2 + 8) = (1, 10).

Worked example

Given vectors a = (4, −1) and b = (−2, 5), use vector arithmetic to find 3a − 2b.

Given vectors a = (4
Illustrated card showing Given vectors a = (4
  1. 1Scale a by 3: 3a = (12, −3)
  2. 2Scale b by 2: 2b = (−4, 10)
  3. 3Subtract component-wise: 3a − 2b = (12 − (−4), −3 − 10)
  4. 4Simplify: (16, −13)

Answer: (16, −13)

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students sometimes add the magnitudes of two vectors directly as if they were numbers, ignoring direction, rather than combining their components or using a vector diagram. Always work component-wise or use the triangle/parallelogram rule for addition.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard