A LevelFurther MathsEdexcelFurther Pure 1Revision NotesFurther VectorsFurther VectorsScalar Triple Product

Scalar Triple Product (Edexcel A Level Further Maths: Further Pure 1): Revision Note

Exam code: 9FM0

Mark Curtis

Written by: Mark Curtis

Updated on

Scalar triple product

What is the scalar triple product?

  • The scalar triple product of the vectors bold a, bold b and bold c is

    • bold a times open parentheses bold b cross times bold c close parentheses

  • The resulting value is a scalar

    • not a vector

  • Recall that the scalar and vector products are

    • bold a times bold b equals vertical line bold a vertical line vertical line bold b vertical line cos theta equals a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 plus a subscript 3 b subscript 3

    • bold a cross times bold b equals vertical line bold a vertical line vertical line bold b vertical line sin theta bold n with bold hat on top equals open vertical bar table row bold i bold j bold k row cell a subscript 1 end cell cell a subscript 2 end cell cell a subscript 3 end cell row cell b subscript 1 end cell cell b subscript 2 end cell cell b subscript 3 end cell end table close vertical bar equals open parentheses table row cell a subscript 2 b subscript 3 minus blank a subscript 3 b subscript 2 end cell row cell a subscript 3 b subscript 1 minus blank a subscript 1 b subscript 3 end cell row cell a subscript 1 b subscript 2 minus blank a subscript 2 b subscript 1 end cell end table close parentheses

How do I calculate the scalar triple product?

  • To calculate the scalar triple product

    • first work out the vector bold b cross times bold c

      • then 'dot it' with the vector bold a

    • or use the determinant formula

      • bold a times open parentheses bold b cross times bold c close parentheses equals open vertical bar table row cell a subscript 1 end cell cell a subscript 2 end cell cell a subscript 3 end cell row cell b subscript 1 end cell cell b subscript 2 end cell cell b subscript 3 end cell row cell c subscript 1 end cell cell c subscript 2 end cell cell c subscript 3 end cell end table close vertical bar

Examiner Tips and Tricks

The determinant formula for the scalar triple product is given in the formula booklet.

What properties of the scalar triple product do I need to know?

  • Swapping the dot and the cross gives the same result

    • bold a times open parentheses bold b cross times bold c close parentheses equals open parentheses bold a cross times bold b close parentheses times bold c

  • If bold a times open parentheses bold b cross times bold c close parentheses equals 0 then bold a, bold b and bold c are coplanar

    • They all lie in the same plane

  • If any two vectors out of bold a, bold b and bold c are parallel then bold a times open parentheses bold b cross times bold c close parentheses equals 0

    • e.g.

      • bold a times open parentheses bold a cross times bold c close parentheses equals 0

      • bold a times open parentheses bold b cross times bold b close parentheses equals 0

      • bold c times open parentheses bold b cross times bold c close parentheses equals 0

  • Cycling through bold a, bold b and bold c in order gives the same result

    • bold a times open parentheses bold b cross times bold c close parentheses equals bold b times open parentheses bold c cross times bold a close parentheses equals bold c times open parentheses bold a cross times bold b close parentheses

    • but be careful not to swap the order

      • as bold b cross times bold c equals negative bold c cross times bold b

      • so bold a times open parentheses bold b cross times bold c close parentheses equals negative bold a times open parentheses bold c cross times bold b close parentheses

Examiner Tips and Tricks

The cycling property of the scalar triple product is given in the formula booklet.

Worked Example

Given that bold a equals 2 bold i plus 3 bold j minus bold k, bold b equals bold i minus 2 bold j and bold c equals negative 5 bold i plus bold j plus 2 bold k, find bold a times open parentheses bold b cross times bold c close parentheses.

Answer:

Method 1

First work out the vector product bold b cross times bold c

  • e.g. by substituting bold b equals open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses and bold c equals open parentheses table row cell negative 5 end cell row 1 row 2 end table close parentheses into bold b cross times bold c equals open parentheses table row cell b subscript 2 c subscript 3 minus blank b subscript 3 c subscript 2 end cell row cell b subscript 3 c subscript 1 minus blank b subscript 1 c subscript 3 end cell row cell b subscript 1 c subscript 2 minus blank b subscript 2 c subscript 1 end cell end table close parentheses

open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses cross times open parentheses table row cell negative 5 end cell row 1 row 2 end table close parentheses equals open parentheses table row cell open parentheses negative 2 close parentheses cross times 2 minus 0 cross times 1 end cell row cell 0 cross times open parentheses negative 5 close parentheses minus 1 cross times 2 end cell row cell 1 cross times 1 minus open parentheses negative 2 close parentheses cross times open parentheses negative 5 close parentheses end cell end table close parentheses

Simplify

open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses cross times open parentheses table row cell negative 5 end cell row 1 row 2 end table close parentheses equals open parentheses table row cell negative 4 end cell row cell negative 2 end cell row cell negative 9 end cell end table close parentheses

Next substitute bold a equals open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses and bold b cross times bold c equals open parentheses table row cell negative 4 end cell row cell negative 2 end cell row cell negative 9 end cell end table close parentheses into bold a times open parentheses bold b cross times bold c close parentheses

Work out this scalar product

open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses times open parentheses table row cell negative 4 end cell row cell negative 2 end cell row cell negative 9 end cell end table close parentheses equals 2 cross times open parentheses negative 4 close parentheses plus 3 cross times open parentheses negative 2 close parentheses plus open parentheses negative 1 close parentheses cross times open parentheses negative 9 close parentheses

Simplify

bold a times open parentheses bold b cross times bold c close parentheses equals negative 5

Method 2

Substitute bold a equals open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses, bold b equals open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses and bold c equals open parentheses table row cell negative 5 end cell row 1 row 2 end table close parentheses into bold a times open parentheses bold b cross times bold c close parentheses equals open vertical bar table row cell a subscript 1 end cell cell a subscript 2 end cell cell a subscript 3 end cell row cell b subscript 1 end cell cell b subscript 2 end cell cell b subscript 3 end cell row cell c subscript 1 end cell cell c subscript 2 end cell cell c subscript 3 end cell end table close vertical bar

open vertical bar table row 2 3 cell negative 1 end cell row 1 cell negative 2 end cell 0 row cell negative 5 end cell 1 2 end table close vertical bar

Calculate this determinant

  • e.g. along the first row

2 open parentheses open parentheses negative 2 close parentheses cross times 2 minus 1 cross times 0 close parentheses minus 3 open parentheses 1 cross times 2 minus open parentheses negative 5 close parentheses cross times 0 close parentheses minus 1 open parentheses 1 cross times 1 minus open parentheses negative 5 close parentheses cross times open parentheses negative 2 close parentheses close parentheses

Simplify

bold a times open parentheses bold b cross times bold c close parentheses equals negative 5

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Mark Curtis

Author: Mark Curtis

Expertise: Maths Content Creator

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.