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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector PropertiesThe Vector Product

The Vector 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 Vector (Cross) Product

What is the vector product?

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

  • The vector product is perpendicular to both vectors

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

    • This is why it is also called the cross product

  • The direction of the vector product follows the right-hand rule

    • Using you right hand:

      • Point your index finger in the direction of the first vector

      • Point your middle finger in the direction of the second vector

      • The direction of the vector product is given by the direction of your thumb

Right hand showing vector cross product. Index finger points left for vector v, middle down for w, and thumb up for normal vector v×w.
Demonstration of the right-hand rule

How is the vector product calculated?

  • One formula for the vector product is bold italic v cross times bold italic w equals blank open parentheses table row cell v subscript 2 w subscript 3 minus blank v subscript 3 w subscript 2 end cell row cell v subscript 3 w subscript 1 minus blank v subscript 1 w subscript 3 end cell row cell v subscript 1 w subscript 2 minus blank v subscript 2 w subscript 1 end cell end table close parentheses 

    • 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 vector product is bold italic v cross times bold italic w equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar sin invisible function application theta bold italic n

Examiner Tips and Tricks

The first formula is given in the formula booklet under the geometry and trigonometry section. The second formula is not given, however, the formula for the magnitude is given open vertical bar bold italic v cross times bold italic w close vertical bar equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar sin invisible function application theta.

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

  • The vector product is not commutative

    • bold italic v cross times bold italic w not equal to bold italic w cross times bold italic v

  • Changing the order of the vectors reverses the direction of the vctor product

    • bold italic v cross times bold italic w equals negative bold italic w cross times bold italic v

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

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

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

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

  • The vector product between a vector and itself is equal to the zero vector

    • bold italic v cross times bold italic v equals bold 0

  • The vector product of two parallel vectors is equal to the zero vector

    • This is because sin 0 degree equals sin 180 degree equals 0

  • The converse is also true

    • If bold italic v cross times bold italic w equals bold 0 for non-zero vectors

    • Then bold italic v and bold italic w must be parallel

  • The absolute value of the vector product of two perpendicular vectors is equal to the product of their magnitudes

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

      • This is because sin 90 degree equals 1

Examiner Tips and Tricks

Learn the differences between the scalar product and the vector product. The vector product does not follow the rules of multiplication of numbers.

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

Worked Example

Calculate the magnitude of the vector 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 begin mathsize 16px style bold italic w equals 3 bold i minus 2 bold j minus bold k end style using

i) the formula bold italic v cross times bold italic w equals blank open parentheses table row cell v subscript 2 w subscript 3 minus blank v subscript 3 w subscript 2 end cell row cell v subscript 3 w subscript 1 minus blank v subscript 1 w subscript 3 end cell row cell v subscript 1 w subscript 2 minus blank v subscript 2 w subscript 1 end cell end table close parentheses blank,

3-10-4-ib-aa-hl-vector-product-we-solution-1a

ii) the formula , given that the angle between them is 1 radian.

3-10-4-ib-aa-hl-vector-product-we-solution-1b

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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.