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The Scalar (Dot) Product (Cambridge (CIE) A Level Maths): Revision Note

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The Scalar ('Dot') Product

The scalar product is an important link between the algebra of vectors and the trigonometry of vectors. We shall see that the scalar product is somewhat comparable to the operation of multiplication on real numbers.

What is the scalar (dot) product?

  • The scalar product between two vectors a and b is represented by bold a times bold b

    • This is also called the dot product because of the symbol used

  • The scalar product between two vectors bold a equals a subscript 1 bold i plus a subscript 2 bold j plus a subscript 3 bold k and bold b equals b subscript 1 bold i plus b subscript 2 bold j plus b subscript 3 bold k is defined as bold a times bold b equals straight a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 plus a subscript 3 b subscript 3

  • The result of taking the scalar product of two vectors is a real number

    • i.e. a scalar

  • For example,

open parentheses 3 bold i minus bold k close parentheses times open parentheses 2 bold i plus 9 bold j plus bold k close parentheses equals 3 cross times 2 plus 0 cross times 9 plus open parentheses negative 1 close parentheses cross times 1 equals 6 plus 0 minus 1 equals 5

and

open parentheses table row 2 row 7 end table close parentheses times open parentheses table row cell negative 8 end cell row 2 end table close parentheses equals 2 cross times open parentheses negative 8 close parentheses plus 7 cross times 2 equals negative 16 plus 14 equals negative 2

  • The scalar product has some important properties:

    • The order of the vectors doesn’t affect the result:

bold a times bold b equals bold b times bold a

  • In effect we can ‘multiply out’ brackets:

  • This means that we can do many of the same things with vectors as we can do when operating on real numbers – for example,

open parentheses bold a minus bold b close parentheses times open parentheses bold a minus bold b close parentheses equals bold a bold times bold a minus 2 bold a bold times bold b plus bold b bold times bold b

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

bold a bold times bold a equals open vertical bar bold a close vertical bar squared

For example,

 open parentheses table row 2 row 7 end table close parentheses times open parentheses table row 2 row 7 end table close parentheses equals 2 squared plus 7 squared equals 53  and  open vertical bar open parentheses table row 2 row 7 end table close parentheses close vertical bar squared equals 2 squared plus 7 squared equals 53

What is the connection between the scalar product and trigonometry?

  • There is another important method for finding bold a bold times bold b bold spaceinvolving the angle between the two vectors theta:

bold a bold times bold b equals open vertical bar bold a close vertical bar open vertical bar bold b close vertical bar cos space theta

  • Here theta is the angle between the vectors when they are placed ‘base to base’

    • when the vectors are placed so that they begin at the same point

  • This formula can be derived using the cosine rule and expanding open parentheses bold a minus bold b close parentheses times open parentheses bold a minus bold b close parentheses

7-3-3-the-scalar-product

Worked Example

7-3-3-the-scalar-_dot_-product-we-solution

Examiner Tips and Tricks

  • When writing a scalar product, it’s important to write a distinctive dot between the vectors – otherwise your meaning will not be clear.

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