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Inner e dot product

  1. Nov 16, 2013 #1
    A simple question: what is the difference between inner product and dot product?
     
  2. jcsd
  3. Nov 16, 2013 #2

    jgens

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    Gold Member

    The dot product is just a specific inner product on Rn.
     
  4. Nov 16, 2013 #3

    HallsofIvy

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    An "inner product" on a given vector space V, over the complex numbers, is any function that, to any two vectors in U, u and v, assigns the complex number, <u, v> such that
    1) For any vector, v, [itex]<v, v>\ge 0[/itex] and [itex]<v, v>= 0[/itex] if and only if v= 0.
    2) For any vectors, u and v, and any complex number, r, r<u, v>= <ru, v>.
    3) For any vectors, u and v, [itex]<u, v>= \overline{v, u}[/itex].

    (If V is a vector space over the real numbers, <u, v> must be real and <u, v>= <v, u>.)

    The "dot product on Rn" is an inner product and the converse is almost true:
    If we take a basis on the vector space V, consisting of "orthonormal vectors" where "orthogonal" is defined as <u, v>= 0 and "normal" as <v, v>= 1, there is a natural isomorphism from V to Rn, where n is the dimension of V, so we can write u and v as "ordered n-tuples" and the inner product on V is exactly the dot product on Rn.
     
  5. Nov 16, 2013 #4
    I understood. But, by the way, if there is a product between vectors involving the modulus and the sine of the angle formed and can result or a scalar or a vector (exterior product and cross product), so, similarly, no exist a prodcut between vectors involving the modulus and the cossine of the angle formed that could result or a scalar or a vector too?
     
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