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Math Help - Can 2 times the dot product of 2 complex numbers equal to this?

  1. #1
    Senior Member x3bnm's Avatar
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    Can 2 times the dot product of 2 complex numbers equal to this?

    There's a section in Gibson's "Elementary Geometry of Differentiable Curves:An Undergraduate Introduction" book on page 6:

    "Example 1.4: Recall that the component of a vector a in the direction of a unit vector b is the vector (a \cdot b)b. (Example 1.1)
    It is useful to express this in complex notation. Note that for any vectors a,b we have 2(a \cdot b) = a \overline{b} + b \overline{a}: in particular when
    b is a unit vector (i.e. b \overline{b} = \left | b \right |^2 = 1) we have 2(a \cdot b)b = a + \overline{a} b^2."


    What I don't understand is why and how is: 2(a \cdot b) = a \overline{b} + b \overline{a}?

    The author said(given above) and I quote "for any vectors" this is true.

    Can anyone kindly shed light on this and explain a bit why this is so?
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  2. #2
    MHF Contributor

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    Re: Can 2 times the dot product of 2 complex numbers equal to this?

    Quote Originally Posted by x3bnm View Post
    "Example 1.4: Recall that the component of a vector a in the direction of a unit vector b is the vector (a \cdot b)b. (Example 1.1)
    It is useful to express this in complex notation. Note that for any vectors a,b we have 2(a \cdot b) = a \overline{b} + b \overline{a}: in particular when
    b is a unit vector (i.e. b \overline{b} = \left | b \right |^2 = 1) we have 2(a \cdot b)b = a + \overline{a} b^2."
    What I don't understand is why and how is: 2(a \cdot b) = a \overline{b} + b \overline{a}?
    Suppose that a=x+iy~\&~b=u+iv.
    You can show that a\overline{b}+\overline{a}b=2(xu+yv).
    But as vectors a\cdot b=xu+yv.
    Thanks from x3bnm
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    Re: Can 2 times the dot product of 2 complex numbers equal to this?

    We have

    a\overline{b}+\overline{a}b = a\overline{b}+\overline{a\overline{b}} = 2Re(a\overline{b})

    The real part is just

    a_1b_1-a_2(-b_2) = a_1b_1+a_2b_2 = a\cdot b
    Thanks from x3bnm
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    Senior Member x3bnm's Avatar
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    Re: Can 2 times the dot product of 2 complex numbers equal to this?

    Thanks a lot Plato and Vlasev. That makes sense now. Again Thanks.
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