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The Scalar Product (DP IB Analysis & Approaches (AA)): Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

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DP Analysis & Approaches (AA): HLDP Applications & Interpretation (AI): HL

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The Scalar (Dot) Product

What is the scalar product?

  • The scalar product is an operation which takes two vectors and outputs a scalar

  • The scalar product between two vectors bold italic v and bold italic w is denoted bold italic v times bold italic w

    • This is why it is also called the dot product

How is the scalar product calculated?

  • One formula for the scalar product is bold italic v times bold italic w equals blank v subscript 1 w subscript 1 plus blank v subscript 2 w subscript 2 plus blank v subscript 3 w subscript 3

    • begin mathsize 16px style bold italic v equals blank open parentheses fraction numerator v subscript 1 over denominator table row cell v subscript 2 end cell row cell v subscript 3 end cell end table end fraction close parentheses end style

    • begin mathsize 16px style bold italic w equals blank open parentheses fraction numerator w subscript 1 over denominator table row cell w subscript 2 end cell row cell w subscript 3 end cell end table end fraction close parentheses end style

  • Another formula for the scalar product is bold italic v times bold italic w equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar cos space theta

Examiner Tips and Tricks

Both formulas are given in the formula booklet under the geometry and trigonometry section.

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

  • The scalar product is commutative

    • bold italic v times bold italic w equals blank bold italic w times bold italic v

  • The distributive law over addition can be used to expand brackets

    •  bold italic u times open parentheses bold italic v plus bold italic w close parentheses equals bold italic u times bold italic v plus bold italic u times bold italic w

  • The scalar product is associative with respect to multiplication by a scalar

    • open parentheses k bold italic v close parentheses times open parentheses bold italic w close parentheses equals bold italic v times open parentheses k bold italic w close parentheses equals k left parenthesis bold italic v times bold italic w right parenthesis

  • The scalar product between a vector and itself is equal to the square of its magnitude

    • bold italic v times bold italic v equals open vertical bar bold italic v close vertical bar squared

  • The absolute value of the scalar product of two parallel vectors is equal to the product of their magnitudes

    • vertical line bold italic v times bold italic w vertical line equals vertical line bold italic w vertical line vertical line bold italic v vertical line

      • This is because cos 0 degree equals 1 and cos space 180 degree equals negative 1

  • The scalar product of two perpendicular vectors is equal to zero

    • This is because cos space 90 degree equals 0

Examiner Tips and Tricks

The scalar product works very similarly to multiplication of numbers.

You do not need to learn these properties if you are studying the AA HL course.

Worked Example

Calculate the scalar product between the two vectors begin mathsize 16px style bold italic v equals blank open parentheses fraction numerator 2 over denominator table row 0 row cell negative 5 end cell end table end fraction close parentheses blank end styleand bold italic w equals 3 bold j minus 2 bold k minus bold i using:

i) the formula bold italic v times bold italic w equals blank v subscript 1 w subscript 1 plus blank v subscript 2 w subscript 2 plus blank v subscript 3 w subscript 3,

3-9-4-ib-aa-hl-the-scalar-product-we-solution-a

ii) the formula bold italic v times bold italic w equals open vertical bar v close vertical bar open vertical bar w close vertical bar cos invisible function application space theta, given that the angle between the two vectors is 66.6°.

3-9-4-ib-aa-hl-the-scalar-product-we-solution-b

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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

    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.