Showing Revision 3 created 05/24/2016 by Udacity Robot.

1. Title:
   Cross Product - Interactive 3D Graphics
2. Description:

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   For a tilted cylinder we were able to look at and think about what axis to
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   rotate around. However we usually won't be able to do this. If I gave you two
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   arbitrary vectors and said quick what's the axis of rotation? Best of luck to
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   you, I sure couldn't do it. Happily there's an easy way to get the axis of
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   rotation and it's called the cross product. In three.js you call it like this,
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   it takes two vectors as its inputs, and the result is put into the vector three
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   itself. The third vector is in fact the axis of rotation or at least one of
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    them. The direction is determined by the right hand rule. You wrap your hand
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    from the first vector, in this case the cylinder to the second vector, in this
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    case the y axis. This then points along the axis of rotation. If we computed the
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    cross product of these two vectors in the opposite order we would go from the y
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    axis to the cylinders vector. And we would get the opposite rotation axis.
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    Recall how the dot product gives us the cosine between two vectors, the length
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    of the cross product result is in fact proportional to the sine of the angle
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    between the two vectors. There is one special case I'm going to point out and
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    its kind of a headache. If the cross product gives back a vector that is of
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    length 0 or nearly so then the two vectors are either pointing in the same
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    direction or in directly opposite directions. You can use the dot product of the
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    two vectors to figure out which, if they point in the same direction, then
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    you're done. You don't need to rotate at all. If they point in exactly opposite
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    directions, then you need to rotate 180 degrees. However, the rotation axis
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    you'll get back from the cross product is actually 0,0,0, which is no axis at
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    all. At this point, you basically need to choose some arbitrary axis that is
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    perpendicular to your vectors and use that for rotation, or just form the
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    rotation matrix directly. Here for example I use the x axis since I know y is
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    going to be perpendicular to it. See the additional course materials on a good
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    way to solve this problem in general. The mathematical notation for a cross
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    product is this, a big X. The length of the vector produced by the cross product
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    is equal to the sign of theta, that's the angle between the two vectors, times
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    the length of A. Times the length of B. The cross product itself is computed by
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    multiplying neighboring elements of the two vectors' coordinates. For the x
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    coordinate, we multiply Ay times Bz and then subtract Az times By. For the y
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    coordinate we multiply Az times Bz minus Ax times Bz. I like to do this kind of
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    x cross thing here as we did before. So I tend to take this and wrap it around
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    to this side. So it's Az times Bx, and Ax times Bz. For the last coordinate we
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    do the same thing, Ax times By minus Ay times Bx. At the end we have a vector
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    that describes the axis of rotation from one vector to the other. And in fact
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    this vector will be perpendicular to both of these two. Oh, and one more thing.
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    If you want to use this vector later you probably going to want to normalize it,
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    because its length will be pretty obscure.