16. The primitive unit vectors a and b of a two dimensional lattice are shown in Figure 2, where n is the unit vector perpendicular to the plane and $\theta$ is the angle between vectors a and b. Show that the magnitude of the reciprocal vector A of vector a is $2\pi/a \sin \theta$. Sketch the direction of A. b n $\theta$ a The real space lattice vectors in 2D. 17. Show that the length of the conventional cell of the reciprocal lattice of fcc is $4\pi/a$. 18. What is the perpendicular separation of adjacent (112) planes? Is this plane denser than the (100) plane? Explain your answer. 19. Construct the 1st, 2nd and 3rd Brillouin zones of a square lattice of lat- tice constant a, indicating how this is done. Mark the high stymmetry points of the 1st Brillouin zone and find their coordinates. 20. For the fcc crystal lattice show that $\Omega_r = 8\pi^3/\Omega_k$
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1.02 Ha Interactive for Chapter Cartesian Vectors Learning Goal: To understand Cartesian representations of vectors and how they can be determined from direction and magnitude; to learn how to represent vectors expressed as the product of the component magnitude and the unit direction vectors i, j, and k; and to learn how to represent a vector component as a function of the angle between the vector and the coordinate axis. Part A: Calculating the magnitude of a vector from its components What is the magnitude of F? Express your answer to three significant figures and include the appropriate units. As shown, a pole is subjected to three forces: F, P, and T. The force expressed in Cartesian vector form is F = [5 i + 10 j + 10 k] N. Force P has a magnitude of 7.55 N and acts in the direction given by the direction angles α = 22.0°, β = 105°, and γ = 74.6° to the x, y, and z axes, respectively. Force T lies within the yz plane, its direction is given by the 3-4-5 triangle shown, and its magnitude is 12 N. Part B: Components of a vector Determine the components of P in the i, j, and k directions. Express your answers, separated by commas, to three significant figures. Part C: Finding Cartesian components from right triangles What are the Cartesian components of force T? Express your answers, separated by commas, to three significant figures.
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