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Basics of Vectors -- Parallel versus Co-Planar

  1. Jun 9, 2017 #1
    1. The problem statement, all variables and given/known data
    Ex. 2. Express each of the following by an equation:
    (a) Two vectors A and B are parallel.
    (b) Three vectors A, B and C are co-planar.

    2. Relevant equations
    C = A + B

    3. The attempt at a solution

    I understand (a) the answer being B = (alpha)*A because that is a scalar transformation, but for part (b) the answer in the end of the book is C = (alpha)*A + (beta)*B where (alpha) and (beta) are both scalar quantities. Wouldn't it still be correct to say that C = A + B without the scalar transformation. I just don't understand why the book chose to use this answer. Thank you for any help.
     
  2. jcsd
  3. Jun 9, 2017 #2

    robphy

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    Let ##\vec A## be the unit vector ##\hat x##.
    Let ##\vec B## be the unit vector ##\hat y##.
    Consider any vector ##\vec C## in the xy-plane. Is ##\vec C=\vec A+\vec B##?
     
  4. Jun 9, 2017 #3

    scottdave

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    While C = A + B does describe 3 vectors which will be coplanar, there are many other A B C combinations, where they do not form a triangle.
    For example, let C = A + B then what can we say about the vector D = A + C? This vector also lies in the same plane, but can you get B from adding these (and not use scalar multipliers)?
    We know that C = A + B, substitute that in, and get D = A + (A + B) = 2A + B. This is why you need the scalar multipliers. Take any two non-parallel vectors in a plane, and you can add scalar multiples of them to produce a new 'triangle' which lies in the plane and the 3rd leg lies in the same plane. I hope this helps.
    Take the special case of vectors i and j, which happen to be length 1 and orthogonal. You can scale each of those and add to get any vector in the i-j plane. But your two vectors, used to build other vectors, do not have to be orthogonal for it to work (but they cannot be parallel).
     
  5. Jun 10, 2017 #4

    haruspex

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    Amusingly, that is not the answer either. If A and B are parallel then the three are necessarily coplanar, but you would only be able to write C as a linear combination of A and B if all three are parallel.

    A correct answer is that there exist three scalars, a, b and c, not all zero, such that ##a\vec A+b\vec B+ c\vec C=0##.

    Technically, there is the same problem with a). A better answer is, likewise, that there exist two scalars, not both zero....
     
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