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Dot product

  1. Oct 9, 2015 #1
    Prop: Suppose a and b be vectors in R3. If a·x=b·x for all vector x in R3, then a=b

    My question if the proposition is always true.
    And if x is a zero vector, is the proposition still valid?
     
  2. jcsd
  3. Oct 9, 2015 #2

    andrewkirk

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    Note the words 'for all vectors x in R3', in the statement. Not 'for some vector x in R3'.

    To see whether it's true for all pairs a,b, consider what you can conclude if it's true for all three of the basis vectors (1 0 0), (0 1 0), (0 0 1).
     
  4. Oct 9, 2015 #3
    Aha, i have missed something. Thanks for ur help@@
     
  5. Oct 9, 2015 #4

    jbunniii

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    Another proof: ##a \cdot x = b \cdot x## if and only if ##(a-b) \cdot x = 0##. If this holds for all ##x##, then choosing ##x = a-b## implies that ##(a-b)\cdot (a-b) = 0##. Since ##(a-b)\cdot (a-b)## is the square of the length of ##a-b##, this means that ##a-b## has length zero, so ##a-b = 0##, and therefore ##a=b##.
     
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