IBMaths: AA HLDPRevision Notes3. Geometry & Trigonometry3.9 Vector Properties3.9.5 The Vector Product

The Vector Product (DP IB Maths: AA HL)

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The scalar product is one method of multiplying vectors which results in a scalar and has uses when working with vectors and lines. The vector product is a different method, which results in a vector and has uses when working with lines and planes.

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The Vector ('Cross') Product

What is the vector (cross) product?

  • The vector product (also known as the cross product) is a form in which two vectors can be combined together
  • The vector product between two vectors v and w is denoted v × w
  • The result of taking the vector product of two vectors is a vector
  • The vector product is a vector in a plane that is perpendicular to the two vectors from which it was calculated
    • This could be in either direction, depending on the angle between the two vectors
    • The right-hand rule helps you see which direction the vector product goes in
      • By pointing your index finger and your middle finger in the direction of the two vectors your thumb will automatically go in the direction of the vector product

3-10-4-ib-aa-hl-right-hand-rule-diagram-1

How do I find the vector (cross) product?

  • There are two methods for calculating the vector product
  • The vector product of the two vectors v and w can be written in component form as follows:
    • begin mathsize 16px style bold italic v cross times bold italic w equals blank open parentheses table row cell bold italic v subscript 2 bold italic w subscript 3 minus blank bold italic v subscript 3 bold italic w subscript 2 end cell row cell bold italic v subscript 3 bold italic w subscript 1 minus blank bold italic v subscript 1 bold italic w subscript 3 end cell row cell bold italic v subscript 1 bold italic w subscript 2 minus blank bold italic v subscript 2 bold italic w subscript 1 end cell end table close parentheses end style 
    • Where 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 and 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
    • This is given in the formula booklet
  • The vector product can also be found in terms of its magnitude and direction
  • The magnitude of the vector product is equal to the product of the magnitudes of the two vectors and the sine of the angle between them
    • vertical line bold italic v cross times bold italic w vertical line equals open vertical bar v close vertical bar open vertical bar w close vertical bar sin invisible function application theta
    • Where θ is the angle between v and w
      • The two vectors v and w are joined at the start and pointing away from each other
    • This is given in the formula booklet
  • The direction of the vector product is perpendicular to both v and w

 

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

  • The order of the vectors is important and changes the result of the vector product
    • bold italic v cross times bold italic w not equal to blank bold italic w cross times bold italic v
    • However
    • bold italic v cross times bold italic w equals blank minus bold italic w cross times bold italic v
  • The distributive law can be used to ‘expand brackets’
    •  bold italic u cross times open parentheses bold italic v blank plus bold italic w close parentheses equals blank bold italic u cross times bold italic v plus blank bold italic u cross times bold italic w
    • Where u, v and w are all vectors
  • Multiplying a scalar by a vector gives the result:
    • open parentheses k bold italic v close parentheses cross times bold italic w equals blank 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 zero
    • bold italic v cross times bold italic v equals 0
  • If two vectors are parallel then the vector product is zero
    • This is because sin 0° = sin 180° = 0
  • If bold italic v cross times bold italic w equals 0 then v and w are parallel if they are non-zero
  • If two vectors, v and w, are perpendicular then the magnitude of the vector product is equal to the product of the magnitudes of the vectors
    • vertical line bold italic v cross times bold italic w vertical line equals vertical line bold italic w vertical line vertical line bold italic v vertical line
    • This is because sin 90° = 1

Exam Tip

  • The formulae for the vector product are given in the formula booklet, make sure you use them as this is an easy formula to get wrong
  • The properties of the vector product are not given in the formula booklet, however they are important and it is likely that you will need to recall them in your exam so be sure to commit them to memory

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 begin mathsize 16px style bold italic v cross times bold italic w equals blank open parentheses table row cell bold italic v subscript 2 bold italic w subscript 3 minus blank bold italic v subscript 3 bold italic w subscript 2 end cell row cell bold italic v subscript 3 bold italic w subscript 1 minus blank bold italic v subscript 1 bold italic w subscript 3 end cell row cell bold italic v subscript 1 bold italic w subscript 2 minus blank bold italic v subscript 2 bold italic w subscript 1 end cell end table close parentheses blank end style,

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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Areas using Vector Product

How do I use the vector product to find the area of a parallelogram?

  • The area of the parallelogram with two adjacent sides formed by the vectors v and w is equal to the magnitude of the vector product of two vectors v and w
    • A equals open vertical bar bold italic v blank cross times blank bold italic w close vertical bar blankwhere v and w form two adjacent sides of the parallelogram
      • This is given in the formula booklet

 

How do I use the vector product to find the area of a triangle?

  • The area of the triangle with two sides formed by the vectors v and w is equal to half of the magnitude of the vector product of two vectors v and w
    • A equals 1 half open vertical bar bold italic v blank cross times blank bold italic w close vertical bar blankwhere v and w form two sides of the triangle
      • This is not given in the formula booklet

Exam Tip

  • The formula for the area of the parallelogram is given in the formula booklet but the formula for the area of a triangle is not
    • Remember that the area of a triangle is half the area of a parallelogram

Worked example

Find the area of the triangle enclosed by the coordinates (1, 0, 5), (3, -1, 2) and (2, 0, -1).

3-10-4-ib-aa-hl-area-we-solution

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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.