Use your protractor to measure an angle of 50°.
Fold the blue pipe cleaner in half, opening the legs so that the angle between
them is 50°. Take the "V" end of the folded blue pipe cleaner and
hold it at the origin. Take one leg of this "V" and fasten it around the
red query pipe cleaner. Make sure the legs of the blue pipe cleaner stay
spread 50°.
Rotate the free, unattached leg of the blue pipe cleaner around the red
pipe cleaner, sweeping out an invisible cone.
Figure 2 shows a visible representation of the invisible
cone. Only the red and blue pipe cleaners, not the green ones, are depicted,
to simplify the picture.
Figure 2: Cone created by sweeping blue pipe cleaner around red pipe cleaner
Which document
vectors fall in this cone? Notice that an angle of 50° has
a cosine measure of 0.643. This means that all document
vectors whose cosines of their angles with the query vector are greater than
0.643 will be retrieved
by your physical 3-D search engine system. Suppose you want to be more
lenient and return to the user all documents with a relevancy score
above 0.3. What angle should you now sweep out? [You desire the angle θ such that cos(θ) = 0.3, which means
θ = cos^{-1}(0.3) = arccos(0.3) = 72.54°.] Now which document vectors would your
physical 3-D search engine return to the user?