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IBMaths: AA HLDPRevision Notes3. Geometry & Trigonometry3.9 Vector Properties3.9.4 The Scalar Product

The Scalar Product (DP IB Maths: AA HL)

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The Scalar ('Dot') Product

What is the scalar product?

  • The scalar product (also known as the dot product) is one form in which two vectors can be combined together
  • The scalar product between two vectors a and b is denoted bold a times bold b
  • The result of taking the scalar product of two vectors is a real number
    • i.e. a scalar
  • The scalar product of two vectors gives information about the angle between the two vectors
    • If the scalar product is positive then the angle between the two vectors is acute (less than 90°) 
    • If the scalar product is negative then the angle between the two vectors is obtuse (between 90° and 180°) 
    • If the scalar product is zero then the angle between the two vectors is 90° (the two vectors are perpendicular)

How is the scalar product calculated?

  • There are two methods for calculating the scalar product
  • The most common method used to find the scalar product between the two vectors v and w is to find the sum of the product of each component in the two vectors
    • 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
    • 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 scalar product is also equal to the product of the magnitudes of the two vectors and the cosine of the angle between them
    • bold italic v times bold italic w equals open vertical bar v close vertical bar open vertical bar w close vertical bar cos space 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
  • The scalar product can be used in the second formula to find the angle between the two vectors

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

  • The order of the vectors doesn’t change the result of the scalar product (it 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 blank plus bold italic w close parentheses equals blank bold italic u times bold italic v plus blank 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 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
  • If two vectors, v and w, are parallel then the magnitude of the scalar product is equal to the product of the magnitudes of the vectors
    • 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° = 1 and cos 180° = -1
  • If two vectors are perpendicular the scalar product is zero
    • This is because cos 90° = 0

Examiner Tip

  • Whilst the formulae for the scalar product are given in the formula booklet, the properties of the scalar product are not, 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 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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Angle Between Two Vectors

How do I find the angle between two vectors?

  • If two vectors with different directions are placed at the same starting position, they will form an angle between them
  • The two formulae for the scalar product can be used together to find this angle
    • begin mathsize 16px style cos space invisible function application theta blank equals blank fraction numerator v subscript 1 w subscript 1 plus blank v subscript 2 w subscript 2 plus blank v subscript 3 w subscript 3 over denominator open vertical bar bold italic v close vertical bar vertical line bold italic w vertical line end fraction end style
    • This is given in the formula booklet
  • To find the angle between two vectors:
    • Calculate the scalar product between them
    • Calculate the magnitude of each vector
    • Use the formula to find cos θ
    • Use inverse trig to find θ

Examiner Tip

  • The formula for this is given in the formula booklet so you do not need to remember it but make sure that you can find it quickly and easily in your exam

Worked example

Calculate the angle formed by the two vectors begin mathsize 16px style bold italic v equals blank open parentheses fraction numerator negative 1 over denominator table row 3 row 2 end table end fraction close parentheses blank end styleand bold italic w equals 3 bold i plus 4 bold j minus bold k.

3-9-4-ib-aa-hl-angle-between-two-vectors-we-solution-a

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Perpendicular Vectors

How do I know if two vectors are perpendicular?

  • If the scalar product of two (non-zero) vectors is zero then they are perpendicular
    • If bold italic v times bold italic w equals 0 then v and w must be perpendicular to each other
  • Two vectors are perpendicular if their scalar product is zero
    • The value of cos θ = 0 therefore |v||w|cos θ = 0

 

Worked example

Find the value of t such that the two vectors begin mathsize 16px style bold italic v equals blank open parentheses fraction numerator 2 over denominator table row t row 5 end table end fraction close parentheses blank end styleand bold italic w equals left parenthesis t minus 1 right parenthesis bold i minus bold j plus bold k are perpendicular to each other.

3-9-4-ib-aa-hl-the-angle-between-vectors-we-solution

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    Amber

    Author: Amber

    Expertise: Maths

    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.