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Component of a vector along another vector.

  1. Oct 22, 2014 #1
    1. The problem statement, all variables and given/known data
    Given ##\vec{A}=2\hat{i}+3\hat{j}## and ##\vec{B}=\hat{i}+\hat{j}##.Find the component of ##\vec{A}## along ##\vec{B}##.

    2. Relevant equations
    ##\vec{A}.\vec{B}=ABcosθ## where θ is the angle between both the vectors.

    3. The attempt at a solution
    I attempted the question as follows:
    Let the angle between ##\vec{A}## and ##\vec{B}## be 'θ'. So the component of ##\vec{A}## along ##\vec{B}## is given by ##Acosθ\hat{B}## => ##Acosθ(\frac{\vec{B}}{B})##

    As ##\vec{A}.\vec{B}=ABcosθ## => ##[( 2\hat{i}+3\hat{j})(\hat{i}+\hat{j})]/B=Acosθ## => ##\frac{5}{\sqrt{2}}=Acosθ##

    Therefore the component is : ##\frac{5}{\sqrt{2}}(\frac{\hat{i}+\hat{j}}{\sqrt{2}})## => ##\frac{5}{2}({\hat{i}+\hat{j}})##

    But my text produces the solution as follows:
    ##A_B=(\vec{A}.\vec{B})\hat{B}=\frac{5}{\sqrt{2}}(\hat{i}+\hat{j})##.
     
  2. jcsd
  3. Oct 22, 2014 #2

    RUber

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    Homework Helper

    I usually see this process broken down into basis components.
    That is ##\hat B =\sqrt{2}/2 \hat i + \sqrt{2}/2 \hat j. ##
    Then the component is ##A\cdot \hat B_i \hat i + A\cdot \hat B_j \hat j ##.
    Somewhere in your process, you divided by the magnitude of B twice.
     
  4. Oct 23, 2014 #3
    You're right and the book is wrong. The book answer as well as the formula for AB they use.
    The length of your answer is smaller than the length of A as it should be. The book answer is larger.
    The projection of A on B should only depend on the direction of B, not the magnitude. The formula used for AB in the book does depend on the magnitude of B.
     
  5. Oct 23, 2014 #4

    Delta²

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    Gold Member

    Book is wrong . We can verify this by standard euclidean geometry easily because by the definition of cosine, it will be cos(θ)=(component of A along B)/A hence Acos(θ)=(component of A along B). And we have to multiply this by the unit vector of B to get the required result.
     
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