National 5MathsSQARevision NotesGeometric SkillsVectorsVector Arithmetic

Vector Arithmetic (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Adding & subtracting vector components

How do I add and subtract vectors in component form?

  • Adding and subtracting vectors is done by looking at the components of the vectors separately

  • To add vectors in component form

    • Add the x components together

    • Add the y components together

    • Add the z components together (only if the vector is three-dimensional!)

      • 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

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

  • To subtract vectors in component form

    • Subtract the second x component from the first

    • Subtract the second y component from the first

    • Subtract the second z component from the first (only if the vector is three-dimensional!)

      • 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

      • open parentheses table row 2 row cell negative 3 end cell row 5 end table close parentheses minus open parentheses table row 1 row 1 row cell negative 4 end cell end table close parentheses equals open parentheses table row cell 2 minus 1 end cell row cell negative 3 minus 1 end cell row cell 5 minus open parentheses negative 4 close parentheses end cell end table close parentheses equals open parentheses table row 1 row cell negative 4 end cell row 9 end table close parentheses

Multiplying a vector by a scalar

How do I multiply a vector by a scalar?

  • A scalar is a regular number not a vector

    • It does not have a direction

  • To multiply a column vector by a scalar

    • Multiply the x component by the scalar

    • Multiply the y component by the scalar

    • Multiply the z component by the scalar (only if the vector is three-dimensional!)

      • 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

      • 4 open parentheses table row 3 row 2 row cell negative 5 end cell end table close parentheses equals open parentheses table row cell 4 cross times 3 end cell row cell 4 cross times 2 end cell row cell 4 cross times open parentheses negative 5 close parentheses end cell end table close parentheses equals open parentheses table row 12 row 8 row cell negative 20 end cell end table close parentheses

How do I write an expression as a single vector in component form?

  • You need to follow the order of operations

    • 2D: space 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

    • 3D: space 3 open parentheses table row cell negative 1 end cell row 5 row 2 end table close parentheses minus 7 open parentheses table row 2 row cell negative 2 end cell row 3 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

    • open parentheses table row cell 3 cross times negative 1 end cell row cell 3 cross times 5 end cell row cell 3 cross times 2 end cell end table close parentheses minus open parentheses table row cell 7 cross times 2 end cell row cell 7 cross times negative 2 end cell row cell 7 cross times 3 end cell end table close parentheses equals open parentheses table row cell negative 3 end cell row 15 row 6 end table close parentheses minus open parentheses table row 14 row cell negative 14 end cell row 21 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

    • open parentheses table row cell negative 3 minus 14 end cell row cell 15 minus open parentheses negative 14 close parentheses end cell row cell 6 minus 21 end cell end table close parentheses equals open parentheses table row cell negative 17 end cell row 29 row cell negative 15 end cell end table close parentheses

  • The single vector at the end is known as a resultant vector

    • open parentheses table row 25 row cell negative 1 end cell end table close parentheses is the resultant vector for space 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

    • open parentheses table row cell negative 17 end cell row 29 row cell negative 15 end cell end table close parentheses is the resultant vector for space 3 open parentheses table row cell negative 1 end cell row 5 row 2 end table close parentheses minus 7 open parentheses table row 2 row cell negative 2 end cell row 3 end table close parentheses

Worked Example

Given bold a equals open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses and bold b equals open parentheses table row 3 row 5 row 4 end table close parentheses, find the resultant vector 4 bold a plus bold b.

Express your answer in component form.

Answer:

First multiply vector bold a by the scalar in front of it

table row cell 4 bold a plus bold b end cell equals cell 4 open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses plus open parentheses table row 3 row 5 row 4 end table close parentheses end cell row blank equals cell open parentheses table row 8 row cell negative 4 end cell row 12 end table close parentheses plus open parentheses table row 3 row 5 row 4 end table close parentheses end cell end table

Then add the components together

table row blank equals cell open parentheses table row cell 8 plus 3 end cell row cell negative 4 plus 5 end cell row cell 12 plus 4 end cell end table close parentheses end cell row blank equals cell open parentheses table row 11 row 1 row 16 end table close parentheses end cell end table

That is the resultant vector in component form

table row blank blank cell open parentheses table row 11 row 1 row 16 end table close parentheses end cell end table

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 space p and the value of q.

Answer:

Write the left-side side as one vector

First multiply each vector by the scalar in front of it

table row cell 2 open parentheses table row p row 3 end table close parentheses plus 3 open parentheses table row cell negative 2 end cell row 1 end table close parentheses end cell equals cell open parentheses table row 4 row q end table close parentheses end cell row cell 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 end cell equals cell open parentheses table row 4 row q end table close parentheses end cell row blank blank blank end table

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 x 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 y components are equal

9 equals q

space p equals 5 space space and space space q equals 9

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.