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Evaluating magnitude of vector

  1. Apr 21, 2015 #1
    1. The problem statement, all variables and given/known data

    Let a and b be two unit vectors such that | a + b| = √3. If
    c = a+ 2b + 3(a x b), then 2|c| is equal to:
    √55
    √51
    √43
    √37
    2. Relevant equations
    a x b = |a| |b| sinθ n where n is a unit vector
    ## | a + b | = \sqrt{a^2 + b^2 + 2abcosθ} ##

    3. The attempt at a solution
    Found cosθ = 1/2 and θ = π/3
    Then 3 a x b = 3√3/2 n
    Now how to find c?
    We don't know angle between n and a or n and b
     
  2. jcsd
  3. Apr 21, 2015 #2

    Ray Vickson

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    Yes, you do know the angle. Go back and review the definition and properties of axb.
     
  4. Apr 21, 2015 #3
    I know angle is π/3
    What to do next?
     
  5. Apr 21, 2015 #4
    I have also written value of a x b in attempt in post 1
     
  6. Apr 21, 2015 #5

    Ray Vickson

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    No. The angle between a and b is π/3, but that is not the angle you asked about. You asked about the angle between a and axb or between b and axb.
     
  7. Apr 21, 2015 #6
    Oh that is 90°. But how I will evaluate c?
     
  8. Apr 21, 2015 #7

    Ray Vickson

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    What is preventing you from writing out all the terms of |c|^2 and evaluating them one-by-one? In other words, use the fact that
    $$ |\vec{c}|^2 = \vec{c} \cdot \vec{c} \\
    = (\vec{a} + 2\vec{b} +3\, \vec{a} \times \vec{b}) \cdot (\vec{a} + 2\vec{b} +3\, \vec{a} \times \vec{b}) $$
    and just expand it all out.
     
    Last edited: Apr 21, 2015
  9. Apr 21, 2015 #8
    ## |c|^2 = (a+2b)^2 + [3( a X b )]^2 + 2( a+2b)3( a X b ) cos θ ##
    What is the angle between (a+ 2b) and a x b ?
    Between a and a x b it is 90°.
     
  10. Apr 21, 2015 #9

    haruspex

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    what does the dot product of the two yield?
     
  11. Apr 21, 2015 #10
    It yields 0, so the angle is 90°?
     
  12. Apr 21, 2015 #11

    haruspex

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    Yes.
     
  13. Apr 21, 2015 #12

    Ray Vickson

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    You have not "expanded it all out"; you should be getting 6 terms, not just the 3 you have written.
     
  14. Apr 21, 2015 #13
    Got it on solving.
    Thanks to both of you.
     
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