Show that the distance measure defined as the angle between two data vec- tors, x and y, satisfies the metric axioms given on page 70. Specifically, d(x, y) = arccos(cos(x, y)).
What will be an ideal response?
Note that angles are in the range 0 to 180?.
1(a). Because 0 ? cos(x, y) ? 1, d(x, y) ? 0.
1(b). Because cos(x, x)=1, d(x, x) = arccos(1) = 0
2. Because cos(x, y) = cos(y, x), d(x, y) = d(y, x)
3. If the three vectors lie in a plane then it is obvious that the angle between
x and z must be less than or equal to the sum of the angles between x and
y and y and z. If y is the projection of y into the plane defined by x and
z, then note that the angles between x and y and y and z are greater than
those between x and y and y and z.Note that angles are in the range 0 to 180?.
1(a). Because 0 ? cos(x, y) ? 1, d(x, y) ? 0.
1(b). Because cos(x, x)=1, d(x, x) = arccos(1) = 0
2. Because cos(x, y) = cos(y, x), d(x, y) = d(y, x)
3. If the three vectors lie in a plane then it is obvious that the angle between
x and z must be less than or equal to the sum of the angles between x and
y and y and z. If y is the projection of y into the plane defined by x and
z, then note that the angles between x and y and y and z are greater than
those between x and y and y and z.
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