document.write( "Question 832848: In a three dimensional x, y, z coordinate system, how many lines through the origin make angles of 60 degrees with both the y and z axes? Also, what angles do they make with the positive x-axis? (My solution manual says that there are only two such lines. I conclude that there are four. Also, it says that the remaining angles with the positive x-axis are 45 and 135 degrees. I cannot see this in three space. Please explain. Thanks.)
\n" ); document.write( "Merc Boyd \n" ); document.write( "

Algebra.Com's Answer #502217 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
There are 4 unique lines through the origin, not 2, so you are correct.
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\n" ); document.write( "3D rotations are difficult to conceptualize, the best way to do it is to use a pencil and play around.
\n" ); document.write( "Think of the pencil as having a length of 1.
\n" ); document.write( "When you rotate it 60 degrees in the xy plane, the point is now at (1/2,sqrt(3)/2,0)
\n" ); document.write( "When you rotate it 60 degrees in the xz plane, the point is now at (1/2, 0, sqrt(3)/2).
\n" ); document.write( "So when you look at it in the yz plane, it has components that run along (sqrt(3)/2,sqrt(3)/2) line which would be 45 degrees. \n" ); document.write( "

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