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Limit question involving vectors

  1. Sep 28, 2011 #1
    A and B are 2 non-zero constant vectors and |B| =1. If the angle between them is pi/4, find the limit of (|A+xB|-|A|)/x as x approaches 0.

    I'm stuck at this question. I've tried using dot product and vector product. But...I don't see the connection ... Any useful hints would be helpful.

  2. jcsd
  3. Sep 28, 2011 #2


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    It might pay to draw a picture of the vectors A and A + xB ; you can make a triangle, the third leg of which is xB . You can then get a relationship between the length of A and of A + xB from the Law of Cosines. Set up your limit expression and see what happens...
  4. Sep 28, 2011 #3
  5. Sep 28, 2011 #4


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    You're trying to work with vector quantities when the Law of Cosines will let you have scalars. We have

    | A + xB |2 = A2 + x2 - 2Ax ( -√2 / 2 ) = A2 + x2 + (√2) Ax

    (we've removed B , since it is a unit vector). The limit we want to evaluate is

    [tex]lim_{x \rightarrow 0} \frac{\sqrt{A^{2} + x^{2} + (√2 \cdot Ax) } - A}{x} .[/tex]

    L'Hopital isn't gonna do it for this because of the radical, so you want to use "conjugate factors" , which will ultimately let you eliminate the "x" in the denominator and have a ratio the limit of which is not indeterminate.
  6. Sep 29, 2011 #5
    Got it!

    You're right. When in doubt,draw.

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