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Homework Help: Orthogonality, point on line closest to point in space

  1. Apr 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the point on the line y = 2x+1 that is closest to the point (5,2)

    2. Relevant equations
    Vector Projection
    x and y are orthogonal (angle between them 90 degrees) if:
    x dot y = 0

    3. The attempt at a solution
    There's a similar example in my book, but it has information that I'm missing. It gives an additional vector in the direction of the line. Then, in order to find the point on the line that's closest to the point given, he just takes the vector projection of v (which would be (5,2) in my case) onto w (vector in the direction of the line).

    This seems relatively straightforward to me, I just don't know how to obtain the value for the vector in the direction of the line. Any guidance? I'd appreciate it!

  2. jcsd
  3. Apr 8, 2009 #2


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    Last edited by a moderator: Apr 24, 2017
  4. Apr 8, 2009 #3


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    If it were me, I wouldn't worry about "vectors" or "projections". The slope of the given line, y= 2x+ 1, is 2. The slope of any line perpendicular to that is -1/2. What is the equation of a line with slope -1/2 through (5, 2)? Where does that line intersect y= 2x+ 1?
  5. Apr 8, 2009 #4
    Oh, wow, duh. I guess I let the language get the best of me. Thank you!
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