Find the orthogonal projection of v onto the subspace w spanned by the vectors ui. (you may assume that the vectors ui are orthogonal.) v = 6 −3 9 −6 , u1 = 1 1 0 0 , u2 = 1 −1 −1 1 , "u3" = 0 0 1 1

Acoal-fired plant generates 600 mw of electric power. the plant uses 4.8 x 10^6 kg of coal each day, and the heat of combustion of coal is 3.3x 10^7 j/kg. the steam that drives the turbines is at a temperature of 300° c, and the exhaust water is at 37° c. (a) what is the overall efficiency of the plant for generating electric power? (b) how much thermal energy is exhausted each day? (c) what is the change in entropy of the universe for this heat engine? (d) using the same heat reservoirs, what is the maximum possible (carnot) efficiency for a heat engine?