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Vectors and Scalars

  1. Oct 2, 2004 #1
    vector U is an unkown vector; known vectors are vectorA and vectorB and scalarF. vectorA (cross) vectorU = vectorB and vectorA (dot) vectorU = F. Express vectorU in terms of vectorA, vectorB, scalarF and the magnitude of vectorA.

    Thank you
     
  2. jcsd
  3. Oct 2, 2004 #2

    mathwonk

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    given A,B,f, such that AxU = B, and A.U = f, find U in terms of A,B,f.

    well lets see what we have. AxU = B means that B is perpendicular to the plane spanned by A and U. A.U = f, means that |A||U|cos(t) = f where t is the angle between A and U. Then B/|B| x A/|A| = C is a unit vector perpendicular to the plane spanned by A and B, hence lies in the plane spanned by A and U. So U is in the plane spanned by A and C, which are perpendicular to each other, but the angle t between A and U has cosine equal to f/(|A||U|. So C and A/|A| =V are orthonormal vectors and U is a linear combination of them. Indeed U/|U| = cos(t)A + sin(t)C.

    now all we need is the length of U. But |B| = |A||U| sin(t), so |U| = |B|/|A|sin(t).

    that is pretty close if you note that sin(t) is positive, hence determined by cos(t).

    does that help?
     
  4. Oct 2, 2004 #3
    Thank you for your reply,

    But could you elaborate on, "that is pretty close if you note that sin(t) is positive, hence determined by cos(t)"

    Thank you for your help
     
  5. Oct 2, 2004 #4

    mathwonk

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    i mean sin(t) = +sqrt(1-cos^2(t)), so everything is known.
     
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