The melakha of Meĥatekh is instrumental in transforming raw material into houses, implements, clothing, etc. For example, if one wishes to make leather clothing, he must first cut the leather to the right size. If one wishes to build a house, he must cut the stones, metal, and panels to the right size. If one wishes to install a window, he must cut the glass to the right size. The general principle is that one who cuts something to a specific size on Shabbat transgresses the Torah prohibition of Meĥatekh. Similarly, it is forbidden by Torah law to cut off the soft part of feathers in order to make pillows and blankets, since one must cut precisely between the hard and soft sections (Shabbat 74b). This is the distinction between Meĥatekh and Kore’a: the primary purpose of Kore’a is to separate two distinct parts, whereas the primary purpose of Meĥatekh is to cut with precision in order to create something new.
The melakha of Meĥatekh does not apply to foods. Therefore one may cut a cake into equal-sized pieces. Similarly, one may cut grooves into an orange in order to peel it. Pills and suppositories used by sick people are also considered food for this purpose, and may be cut where scored (SSK 33:4).
Items that animals would consume are considered food and thus not subject to Meĥatekh. Therefore, one may use a knife to cut straw or hay to use as a toothpick. However, one may not fashion a toothpick from hard wood. Since it is inedible to animals, cutting it to size constitutes Meĥatekh: if by means of a tool, one violates a Torah prohibition, and if by hand, which is considered a shinui, a rabbinic prohibition (Beitza 33a-b; MT 11:7; SA 322:4; MB 322:13, 18).
One may take fragrant branches of hard wood that were harvested before Shabbat and trim or rub them so that their aroma spreads. One may also break them into smaller pieces so that more people can smell them. Even though the branches are hard and animals would not eat them, one may break them, since the size of the pieces does not matter. One may only do so on condition that they are broken by hand. If he uses a tool, we are concerned that he will forget and fashion a toothpick, thus transgressing Torah law (Beitza, loc. cit.; SA 322:5; MB ad loc. 17-18; Rema 336:8).