Because it's a new basis, where the u vector denotes the equivelant of the x direction, the v vector denotes the equivelant of the y direction and the w vector denotes the equivelant of the z direction. So multiplying x by the u vector, y by the v vector and z by the w vector and adding the vectors together will give you another vector which will tell you the coordinates in the new basis when you sub in x y and z coordinates. Think i may have explained quite badly but maybe read up more on basis vectors to understand what they are etc