# Euclidean space - computing distance

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.

© SolutionLibrary Inc. solutionlibary.com 9836dcf9d7 https://solutionlibrary.com/math/calculus-and-analysis/euclidean-space-computing-distance-3bb#### Solution Preview

...od for finding the distances to subspaces.

However we can turn this line into a subspace by a simple change of coordinates.

Let us move the origin to the point v_0. This is equivalent to the change of

coordinates

x_{new} = x_{old} - v_0.

And now let us compute the coordinates of point b in this new coordinate system:

b_0 = b + v_0 = (1, 0, 0, 1)^T - (0, 1, 1, 0)^T = (1, -1, -1, 1)^T.

The new coordinate system is indicated by the dashed lines on the figure.

You will agree with me that in this coordinate system line L is a subspace

since it passes through the origin, and the ...