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    Euclidean space - computing distance

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    Compute the distance from a point b = (1, 0, 0, 1)^T to a line which passes through two points (0, 1, 1, 0)^T and (0, 1, 0, 2)^T. Here ^T denotes the operation of transposition, i.e. the points are represented by column-vectors instead of row-vectors.

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    Solution Preview

    Let us sketch the picture. Obviously we cannot effectively draw vectors in 4-dimensional space, but the dimension in this problem is of little importance. We will sketch all the vectors in two dimensions to understand what's going on (see Attachment 1).

    Let us introduce some notations: we will denote as v_0 the point (0, 1, 1, 0)^T and
    as v the directional vector of the line L that passes through the two given
    points, i.e.

    v = (0, 1, 0, 2)^T - (0, 1, 1, 0)^T = (0, 0, -1, 2)^T;

    Since the line does not pass though the origin, and hence, is not a subspace
    we cannot use the straightforward ...

    Solution Summary

    This shows how to compute the distance from a point to a line with points represented by column vectors instead of row vectors.