Consider two vectors a and b. (12 pts: 3+3+3+3) (a) define the dot product mathematically in two different ways. (hint: one involving components, one involving lengths/angle.) (b) give a brief physical interpretation of what the dot product means. (give a concrete example if you like.) (c) define the vector cross product mathematically in two different ways. (hint: one involving components, one involving lengths/angle.) (d) give a brief physical interpretation of what the cross product means. (give a concrete example if you like.) extra credit (6 pts) find the angle between the "face diagonal" and "body diagonal" of a cube

(a) calculate the number of electrons in a small, electrically neutral silver pin that has a mass of 10.0 g. silver has 47 electrons per atom, and its molar mass is 107.87 g/mol. (b) imagine adding electrons to the pin until the negative charge has the very large value 1.00 mc. how many electrons are added for every 109 electrons already present

Asteel tube (k=50 w/mk) of inner and outer diameter di = 20mm and do=26mm, respectively, is used to transfer heat from hot gases flowing over the tube (h-hot = 200w/m2k) to cold water flowing through the tube (h-cold = 8000 w/m2k). what is the cold-side overall heat transfer coefficient uc? to enhance heat transfer 16 straight fins of rectangular profile are installed longitudinally along the outer surface of the tube. the fins are equally spaced around the circumference of the tube, each having a thickness of 2mm and a length of 15mm. what is the corresponding overall heat transfer coefficient uc? *solve for uc in both cases*

Nthe 19th century, women were expected to conform to a certain role that entailed a. gaining domestic skills and establishing a career b. going to college and getting married c. getting married and having children d. graduating from college and being a housewife