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Math Help - Circles & Vectors

  1. #1
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    Circles & Vectors

    In circle geometry, there is a mathematical property that states a radius, drawn to bisect a chord, will meet the chord at 90 degrees.

    I must prove this property using vector methods. How do I do this? I posted a similar question here some times ago that uses non-vector methods. I did eventually figure that out, but now I need to use vector methods.

    Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by DJ Hobo View Post
    In circle geometry, there is a mathematical property that states a radius, drawn to bisect a chord, will meet the chord at 90 degrees.

    I must prove this property using vector methods. How do I do this? I posted a similar question here some times ago that uses non-vector methods. I did eventually figure that out, but now I need to use vector methods.

    Any help would be appreciated.
    1. Let C denote the center of the circle and the origin.

    Then the vectors \vec a and \vec b are position vectors pointing at points on the circle line. Therefore

    |\vec a| = |\vec b| = r

    2. The vector \overrightarrow{AB} = \vec b - \vec a

    3. The vector \vec m = \frac12(\vec a + \vec b) has the same direction as the line passing through C and M, the midpoint of \overline{AB}

    4. Calculate

    \vec m \cdot \overrightarrow{AB} = \frac12(\vec a + \vec b) \cdot (\vec b - \vec a) = \frac12\left( \vec a \vec b + (\vec b)^2 - (\vec a)^2 - \vec a \vec b \right)

    Since |\vec a| = |\vec b| the value in the bracket is zero. Therefore \vec m and \overrightarrow{AB} are perpendicular.
    Attached Thumbnails Attached Thumbnails Circles & Vectors-circle_vectors.png  
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