Draw a right-handed 3D Cartesian coordinate system (= x, y and z axes). Show a vector A with tail in the origin and sticking out in the positive x, y and z directions. Show the angles between A and the positive x, y and z axes, and call these angles α₁, α₂ and α3 Prove that Ax Acos α₁ Ay = Acos α₂ A₂- Acos α3

solve for Vo

Draw a third quadrant vector C. (remember that boldface characters represent vector quantities). Show the standard angle 0 for this vector (= angle that C makes with the positive x- axis). Also show the angle that C makes with the negative y-axis: call the latter angle 8. Finally, show the smallest angles that C makes with the positive x-axis and the positive y-axis: call these angles p1 and p2, repectively. a) Prove the following formulas for the components of C involving the standard angle (hint: start with the formulas for the components based on the angle & and then use (look up if necessary) co-function identities linking cosine and sine of 8 to sine and cosine of 0 since 8 = 3π/2-8 (this will switch cosine and sine around and eliminate - signs as well)) - C=Ccose C₁=Csine b) Prove the following formulas for the components of C: C=Ccosp1 C=Ccosp2

Notation matters when working with vectors! In particular, it is important to distinguish between the vector itself (A) and its magnitude (A). Illustrate in four separate sketches that each of the following statements is possible: a) both R = A + B and R=A+B are correct b) R = A + B is correct, but R=A+B is incorrect c) R = A + B is incorrect, but R=A+B is correct d) both R = A + B and R=A+B are incorrect

You know from your math courses that an infinitesimal segment of a circular arc can be considered as a straight line segment. Imagine that you cover a full circle in, say, the clockwise direction, with infinitesimal displacement vectors dr. Then evaluate fdr and fdr (the circle symbol on the integral just reminds us that we have to go around the full circle).

When 1.00 g of water at 100˚C changes from the liquid to the gas phase at atmospheric pressure, its change in volume is: 1.67 x 10^-3 How much heat is added to vaporize the water? How much work is done by the water against the atmosphere in expansion? What is the change in the internal energy of the water?

1 m3 of pure water is heated from 10˚C to 120˚C at a constant pressure of 1 atm. The volume of the water is contained, but allowed to expand as needed remaining at 1 atm. Calculate the change in enthalpy of the water. You are provided with the following information at the conditions of 1 atm: The density of pure water between 10˚C and 100˚C: 1000kh/m^3 The heat capacity of water: 4.18 kj/kgK Enthalpy required to convert liquid water to gas (enthalpy of vaporization): 2260 kj/kg The heat capacity of steam: 1.7kj/kgk Is the reaction endothermic or exothermic? Why?

When a dilute gas expands quasi-statically from 0.50 to 4.0 L, it does 250 J of work. Assuming that the gas temperature remains constant at 300 K. What is the change in the internal energy of the gas? How much heat is absorbed by the gas in this process?

A high-speed lifting mechanism supports an 881 kg object with a steel cable that is 22.0 m long and 4.00 cm^2 in cross-sectional area.  Young's modulus for steel is 20.0 ⋅10^10 Pa.  Determine the elongation of the cable.