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Angle between vectors

  1. Jun 16, 2013 #1
    I have some question guys

    i have four points in the x,y plane in cartesian coordinates.

    A (Ax,Ay)
    B(Bx,By)

    C(Cx,Cy)
    D(Dx,Dy)

    A and B is vector G
    C and D is vector F

    I would like to know what is the equation to get the angle between those two vectors (G,F) . and what are the limitations of this equation.

    thank you
     
  2. jcsd
  3. Jun 16, 2013 #2

    Fredrik

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    Are you familiar with the dot product? The angle ##\theta## between two vectors G and F satisfies ##G\cdot F=|G||F|\cos\theta##.

    When we're dealing with vectors in ##\mathbb R^2## or ##\mathbb R^3##, this is a theorem. It holds for all pairs of vectors in those spaces. When we're dealing with an arbitrary inner product space over ##\mathbb R##, we can instead take ##\langle G,F\rangle=\|G\|\|F\|\cos\theta## as the definition of "the angle between G and F". (If you don't know what that means, don't worry about it. These terms are explained in courses on linear algebra). This works because the Cauchy-Schwartz inequality ensures that
    $$\frac{\langle G,F\rangle}{\|G\|\|F\|}\leq 1$$ for all G and F.

    This isn't a textbook-style question, and it doesn't involve physics, so I'm moving it to linear algebra.
     
  4. Jun 16, 2013 #3
    thank you!but..

    it is for excel chart.
    i dont know how to make the vectors for the equation when i use cartesian coordinate.
    i mean how do i make the vector G from A(Ax,Ay) and B(Bx,By)
    can someone write it down for me.

    thank you !!!
     
  5. Jun 16, 2013 #4

    Fredrik

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    ##G=B-A=(B_x,B_y)-(A_x,A_y)=(B_x-A_x,B_y-A_y)##
     
  6. Jun 17, 2013 #5
    sorry again.

    How do i make the last one to the dot product equation?

    the vector G is not a number now but set of coordinates . again sorry but i am a novice and i need full explanation from start to end

    thank you, youre a life saver

    the dude
     
  7. Jun 17, 2013 #6

    Fredrik

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    Is this what you mean?
    $$G\cdot F=(B_x-A_x,B_y-A_y)\cdot (D_x-C_x,D_y-C_y) =(B_x-A_x)(D_x-C_x)+(B_y-A_y)(D_y-C_y)$$ The norm ##\|v\|## of a vector ##v=(x,y)## is defined by ##\|v\|=\sqrt{x^2+y^2}##.
     
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