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Thread: Why Coordinate Axis Orthogonal

  1. #1
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    Why Coordinate Axis Orthogonal

    Not sure if this is geometry or Algebra, but I was wondering, why the axes for cartesian representations are at right angles to each other?
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  2. #2
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    Re: Why Coordinate Axis Orthogonal

    Quote Originally Posted by Atomic_Sheep View Post
    Not sure if this is geometry or Algebra, but I was wondering, why the axes for cartesian representations are at right angles to each other?
    Because orthogonality is useful as a basis for the coordinate system. In higher mathematics, you will find that given any arbitrary representation for a point in Euclidean space, there exists an orthogonal basis which can be used to represent it. This is a powerful result in linear algebra. There exist other coordinate systems such as projective geometries where orthogonality works differently than what you expect. For example, a line can intersect itself in a projective space, and two orthogonal lines can intersect more than once.
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  3. #3
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    Re: Why Coordinate Axis Orthogonal

    A major reason for using orthonormal coordinates is convenience. When writing v in terms of an orthonormal basis, $\displaystyle \{e_1, e_2, ..., e_n\}$, the coefficient of $\displaystyle e_i$ is the dot product of v and $\displaystyle e_i$.
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