Name: _________________________________________
Extra Section 9.4 Practice - OPTIONAL
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1. Consider the vector c = a ´ a ´ b . Is c ^ a ? Is c ^ b ? Either prove or give a
counterexample.
2. Suppose that j ´ a = i . Give two possible solutions for a , and discuss other possible
solutions.
3. Given a = 1, 2, 3 and b = 1,-1,-1 , sketch the collection of all position vectors c satisfying
a ´ b = a ´ c.
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4. Show that a is perpendicular to a - b ´ a + b .
One way to accurately render three-dimensional objects on a computer screen involves using the
dot and cross products. In order to determine how to shade a piece of a surface we need to
determine the angle at which rays from the light source hit the surface. To determine this angle,
we compute the dot product of the light vector with the vector perpendicular to the surface at
the particular point, called the normal vector.
If the light ray hits the surface straight on, that is, has an angle of 180 with the normal, then the
piece of surface will appear bright. On the other hand, if the light comes in on an angle, this piece
of the surface will not appear as bright.
Suppose the light source is placed directly above the xy-plane, so that the light rays come in
parallel to the vector 0, 0, -1 . At what angle (to the normal) do the light rays hit a triangle
bounded by the points
5. (3, 2, 4), (2, 5, 3), and (1, 2, 6)?
6. (3, 5, 2), (3, 3, 1), and (1, 3, 1)?
7. Suppose we are standing above the light source looking down on the xy-plane. Which of these
two regions will appear brighter to us?