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Vector multiplication

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  1. Jun 10, 2015 #1
    Hi i am trying to understand this thoroughly

    Basically i am trying to understand vector multiplication, i dont know if it is the cross product or the dot product i am thinking of

    Okay so here is the question and what is confusing me in the answer

    So if we have two vectors and we multiply them

    a.b this in my mind as i understand it means this(in 2d):

    (a_x + a_y ) * (b_x +b_y) = a_x * b_x + a_x * b_y + a_y * b_x + a_y * b_y

    now i dont understand why the dot product misses these in the centre? and goes straight to only a_x * b_x + a_y * b_y (or the other way a.b.cos(theta))

    what is it exactly that is being multiplied here? and furthermore what exactly is being found here?

    if i wanted to move a vector from position a -> position b , by using the dot product am i finding that space in between? (i.e the vector which is required to be added to a to transform vector a to b?)

    its really confusing me all of this?
     
  2. jcsd
  3. Jun 10, 2015 #2

    PeroK

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    Your confusion is quite fundamental. You probably need to find a good introductory text on vectors and vector operations and study it.

    The dot product shouldn't be hard to understand. It has an algebraic and a geometric significance and has lots of applications in physics.

    A good text book will explain all this.
     
  4. Jun 10, 2015 #3
    can you just explain this to me if you dont mind

    what happens to a_x * b_y + a_y * b_x
     
  5. Jun 10, 2015 #4

    PeroK

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    Nothing happens to them. These terms are simply not part of the dot product.
     
  6. Jun 10, 2015 #5

    A.T.

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    Well, of course you can define new types of vector products, based on what jumps into your mind. What you wrote above is the product of Manhattan norms:
    http://en.wikipedia.org/wiki/Norm_(mathematics)#Taxicab_norm_or_Manhattan_norm
     
    Last edited: Jun 10, 2015
  7. Jun 10, 2015 #6
    okay i have understood this and sorted it out

    this is what i was looking for

    i did confuse the dot product with vector addition

    so if i have a vector


    /|
    / |
    / | 7
    / |
    5

    and i wanted to move this point to say
    /|
    / |
    / |8
    / |
    3

    i would need to add the vector

    /|
    / | 1
    / |
    -2

    and dot product is actually only multiplying a vector which has nothing to do with the components but rather with the complete vector magnitude itself

    such as a . b would be (if the angle between them is 15 degrees/radians)

    a.b cos(15) because you would be getting the component of b which is in line with vector a so that they are both in the same direction and simply multiply them as if they are another scalar * vector multiplication
     
  8. Jun 10, 2015 #7
  9. Jun 10, 2015 #8

    jtbell

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    Your vectors are incomplete because they don't include the unit vectors: $$\vec A = a_x \hat x + a_y \hat y \\ \vec B = b_x \hat x + b_y \hat y$$ The product is $$ \vec A \cdot \vec B = (a_x \hat x + a_y \hat y) \cdot (b_x \hat x + b_y \hat y) \\ \vec A \cdot \vec B = a_x b_x (\hat x \cdot \hat x) + a_x b_y (\hat x \cdot \hat y) + \cdots$$ I'll let you fill in the rest. Some of the dot products of unit vectors equal zero, and some equal 1.
     
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