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I Multiplying a vector by a complex number

  1. Dec 17, 2016 #1
    I have learned that if I multiply a vector, say 3i + 4j, by a scalar that is a real number, say 2, the effect of the operation is to expand the size of the magnitude of the original vector, by 2 in this case, and the result would be 6i + 8j.

    What would be the effect on a vector, like 3i + 4j, if I multiply it by a scalar that is a complex number, say 2 + 5i?
  2. jcsd
  3. Dec 17, 2016 #2


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    If you have a scalar ##z##, be it real or complex, and multiply with a vector ##v = \sum_{i=1}^N v_ie_i## where ##N## the dimension of the vector, ##v_i## the i-th component which can be real or complex scalar, and ##e_i## the i-th basis (unit) vector, the result will be ##zv = \sum_{i=1}^N (zv_i) e_i##.
    Note that there is possible confusion in your notation, you use ##i## both for denoting the component of a vector and the imaginary number ##i=\sqrt{-1}##.
    Last edited: Dec 18, 2016
  4. Dec 17, 2016 #3


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    When you multiply by a complex number you essentially create a dyad of vectors ##(v_1,v_2)## one for the real part (##v_1##) and one for the imaginary part (##v_2##), where in your example it would be ##v_1=2(3i+4j)=6i+8j## and ##v_2=5(3i+4j)=15i+20j##
  5. Dec 18, 2016 #4


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    For this operation to have meaning, you would need to consider your vector to have complex coordinates. So in your vector, the "6" is in reality (6 + 0i) etc.You will find that most vector operations will be the same when using complex coordinates.

    The scalar product (dot product) will have a slightly different definition. Instead of [itex]\vec{a}\cdot \vec{b} =\sum_{k}a_{k}b_{k} [/itex] the definition is [itex] \vec{a}\cdot \vec{b} =\sum_{k}a_{k}\bar{b}_{k}[/itex] where the bar denotes complex conjugation.
  6. Dec 18, 2016 #5


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    It is probably better to rename the parts of your vector: 3x+4y. Then your multiplication gives (6+10i)x + (8+20i)y. Your vector space has to be a vector space over the complex numbers to make such a multiplication meaningful.
  7. Dec 19, 2016 #6
    thank you so much for your quick responses, guys! you have been very helpful.
  8. Dec 19, 2016 #7
    so according to what you have exposed, there is no geometrical interpretation for multiplying a vector (with real numbers as components) by a complex number? performing that kind of operation won't make the vector to have an expanded magnitude, to rotate, or something else?
  9. Dec 19, 2016 #8


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    Not that I know of.

    A vector with real components is an element of a vector space over the field of real numbers. I.e., the components of the vector are real numbers. The basic operations on such a vector space are vector addition and multiplication by scalars (real numbers in this case).

    A vector with complex components is an element of a vector space over the field of complex numbers. The basic operations are the same as above, except that scalar multiplication includes multiplying by complex scalars.
  10. Dec 21, 2016 #9
    Thank you so much for your help!!!!
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