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Introduction to Column Vectors (Cambridge (CIE) IGCSE International Maths)

Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 2Cambridge (CIE) Maths: ExtendedEdexcel Maths A: Higher

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Basic Vectors

What are column vectors?

  • A column vector can be used to describe how to get from one point to another point

    • This is also called a translation vector

    • open parentheses table row 6 row 3 end table close parentheses means 6 units to the right and 3 units up

Column vector

How do I add and subtract column vectors?

  • Adding and subtracting vectors is done by looking at the top numbers and bottom numbers separately

  • To add column vectors

    • Add the top numbers together

    • Add the bottom numbers together

      • open parentheses table row 5 row 2 end table close parentheses plus open parentheses table row 3 row cell negative 1 end cell end table close parentheses equals open parentheses table row cell 5 plus 3 end cell row cell 2 plus open parentheses negative 1 close parentheses end cell end table close parentheses equals open parentheses table row 8 row 1 end table close parentheses

  • To subtract column vectors

    • Subtract the second top number from the first

    • Subtract the second bottom number from the first

      • open parentheses table row 5 row 2 end table close parentheses minus open parentheses table row 3 row cell negative 1 end cell end table close parentheses equals open parentheses table row cell 5 minus 3 end cell row cell 2 minus open parentheses negative 1 close parentheses end cell end table close parentheses equals open parentheses table row 2 row 3 end table close parentheses

How do I multiply a vector by a scalar?

  • A scalar is number not a vector

    • It does not have a direction

  • To multiply a column vector by a scalar

    • Multiply the top number by the scalar

    • Multiply the bottom number by the scalar

      • 3 open parentheses table row 2 row cell negative 1 end cell end table close parentheses equals open parentheses table row cell 3 cross times 2 end cell row cell 3 cross times open parentheses negative 1 close parentheses end cell end table close parentheses equals open parentheses table row 6 row cell negative 3 end cell end table close parentheses

How do I write an expression as a single column vector?

  • You need to follow the order of operations

    • 2 open parentheses table row 5 row 2 end table close parentheses plus 5 open parentheses table row 3 row cell negative 1 end cell end table close parentheses

  • STEP 1
    Multiply each vector by the scalar in front of it

    • open parentheses table row cell 2 cross times 5 end cell row cell 2 cross times 2 end cell end table close parentheses plus open parentheses table row cell 5 cross times 3 end cell row cell 5 cross times open parentheses negative 1 close parentheses end cell end table close parentheses equals open parentheses table row 10 row 4 end table close parentheses plus open parentheses table row 15 row cell negative 5 end cell end table close parentheses

  • STEP 2
    Add or subtract the new column vectors

    • open parentheses table row cell 10 plus 15 end cell row cell 4 plus open parentheses negative 5 close parentheses end cell end table close parentheses equals open parentheses table row 25 row cell negative 1 end cell end table close parentheses

Worked Example

bold a equals open parentheses table row p row 3 end table close parentheses and bold b equals open parentheses table row cell negative 2 end cell row 1 end table close parentheses.

Given that 2 bold a plus 3 bold b equals open parentheses table row 4 row q end table close parentheses, find the value of p and the value of q.

Write the left-side side as one vector
Multiple each vector by the scalar in front of it

open parentheses table row cell 2 p end cell row 6 end table close parentheses plus open parentheses table row cell negative 6 end cell row 3 end table close parentheses equals open parentheses table row 4 row q end table close parentheses

Add the vectors together

open parentheses table row cell 2 p minus 6 end cell row 9 end table close parentheses equals open parentheses table row 4 row q end table close parentheses

The top components are equal
Form and solve an equation

table row cell 2 p minus 6 end cell equals 4 row cell 2 p end cell equals 10 row p equals 5 end table

The bottom components are equal

9 equals q

bold italic p bold equals bold 5 and bold italic q bold equals bold 9

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.