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.)
Merc Boyd
Found 2 solutions by math-vortex, Fombitz:
Answer by math-vortex(648)   (Show Source):
You can put this solution on YOUR website!
Hi, there--
I'd like to know what you have so far.
Describe the four lines you and angles you found. Then we can discuss whether your lines are valid or
perhaps the solution manual is in error (it happens!)
You can post here. If your work is on paper, you may take picture and email it to me at the address
below.
Good luck,
Mrs. F.
math.in.the.vortex@gmail.com
Answer by Fombitz(32388)  (Show Source):
You can put this solution on YOUR website! There are 4 unique lines through the origin, not 2, so you are correct.
.
.
.
3D rotations are difficult to conceptualize, the best way to do it is to use a pencil and play around.
Think of the pencil as having a length of 1.
When you rotate it 60 degrees in the xy plane, the point is now at (1/2,sqrt(3)/2,0)
When you rotate it 60 degrees in the xz plane, the point is now at (1/2, 0, sqrt(3)/2).
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.
|
|
|
