[SOLVED] Physics 311 Homework Set 1

1. By writing vectors in terms of the components, explicitly show that the cross product is
distributive, i.e. Show that
~ (~ + ~)=(~ ~)+(~ ~) (1)2. Find the angle between the face diagonal and a body diagonal of a cube.3. Use the cross product to find the components of the unit vector n perpendicular to the
shaded plane in Fig. 1.11 in your textbook.4. By writing out both sides of the equation in component form, explicitly show that
A~ (B~ C~ ) = B~ (A~ C~ ) C~ (A~ B~ ) (2)5. Find the separation vector ~r from the source point (1,4,5) m to the field point (1,2,3) m.
Determine its magnitude |r|, and construct the unit vector r.6. Find the gradients of the following functions:
a) f(x, y, z) = x2 + y5 + z3.
b) f(x, y, z) = x2y5z3.
c) f(x, y, z) = ln(2x)ez sin(3y).7. The height of a certain hill (in feet) is given by
h(x, y) = 3(2xy 32 4y2 8x + 8y + 5) (3)
where y is the distance (in miles) north, x the distance east of Topeka, Kansas.a) Where is the top of the hill located?b) How high is the hill?c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile east of
Topeka? In what direction is the slope steepest, at that point?