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Thread: vector addition

  1. #1 vector addition 
    Forum Professor wallaby's Avatar
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    perhaps this should go in the maths section, but since i read it out of a physics book i'll put it here.

    i read in a physics book one day you see and came across an equation for addition of vectors basically saying, when vectors A and B are added the resultant vector is equal to:

    R<sup>2</sup>=A<sup>2</sup>+B<sup>2</sup>+2(A)(B)(COS a)

    the first part i can see as being pythagorus and only applying to addition at right angles, so by logical deduction the second part is for addition at non-right angles.

    can someone explain why we add 2(A)(B)(COS a) to obtain the answer?


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  3. #2  
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    Thats a binom, with ab cos(ab) as the scalar product.


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  4. #3  
    Forum Professor wallaby's Avatar
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    why do we add the scalar product and the vector magnitude?
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  5. #4  
    Forum Radioactive Isotope mitchellmckain's Avatar
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    This is known as the "law of cosines" in trigonometry. And yes it is a generalization of the pythagorean theorem to triangles which are not right triangles. Caution, however, a in your formula is not the angle opposite side A, but the angle opposite side R of the triangle. Ultrashotgun is pointing out that it can be derived from vector math.

    Given that R is the vector sum of two vectors A and B, R = A+B, then the magnitude of R is the square root of the dot product of R with itself, or in other words,
    R^2 = R dot R = (A+B) dot (A+B)
    where R is the magnitude of the vector R then by the distributive property we havt that
    R^2 = A dot A + A dot B + B dot A + B dot B
    but A dot A is just A^2 the square of the magnitude of A, likewise, B dot B is B^2.
    and A dot B = B dot A = A*B*cos(angle between vectors A and B)
    where A and B are the magnitudes of vectors A and B.

    So we have
    R^2 = A^2 + B^2 + 2*A*B*cos(angle between vectors A and B)
    See my physics of spaceflight simulator at http://www.relspace.astahost.com

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  6. #5  
    Forum Professor wallaby's Avatar
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    thank you all.

    thats cleared up a lot of confusion for me.
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