How to find the direction cosines of the line which bisects the angle positive degrees between positive...
$begingroup$
I have a text book solution like -
The line bisects angle between Y and Z axes. Therefore the line lies in the YZ plane. Hence the X-axis is perpendicular to the line.
Now how come X-axis is perpendicular to the line in question if the fig. for this problem is like below where X-axis clearly doesn't look perpendicular to the line in question.
linear-algebra geometry
$endgroup$
add a comment |
$begingroup$
I have a text book solution like -
The line bisects angle between Y and Z axes. Therefore the line lies in the YZ plane. Hence the X-axis is perpendicular to the line.
Now how come X-axis is perpendicular to the line in question if the fig. for this problem is like below where X-axis clearly doesn't look perpendicular to the line in question.
linear-algebra geometry
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$begingroup$
Figures are generally not drawn to scale
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– Shubham Johri
Dec 11 '18 at 16:50
$begingroup$
agreed...but there should at least be some coherence to what is being claimed in textbook answer, because if the bisecting line has to be perpendicular to x-axis it cannot exist in yz plane but rather in xy plane..please tell me how i am wrong?
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:55
add a comment |
$begingroup$
I have a text book solution like -
The line bisects angle between Y and Z axes. Therefore the line lies in the YZ plane. Hence the X-axis is perpendicular to the line.
Now how come X-axis is perpendicular to the line in question if the fig. for this problem is like below where X-axis clearly doesn't look perpendicular to the line in question.
linear-algebra geometry
$endgroup$
I have a text book solution like -
The line bisects angle between Y and Z axes. Therefore the line lies in the YZ plane. Hence the X-axis is perpendicular to the line.
Now how come X-axis is perpendicular to the line in question if the fig. for this problem is like below where X-axis clearly doesn't look perpendicular to the line in question.
linear-algebra geometry
linear-algebra geometry
asked Dec 11 '18 at 16:40
Shaikh SakibShaikh Sakib
195
195
$begingroup$
Figures are generally not drawn to scale
$endgroup$
– Shubham Johri
Dec 11 '18 at 16:50
$begingroup$
agreed...but there should at least be some coherence to what is being claimed in textbook answer, because if the bisecting line has to be perpendicular to x-axis it cannot exist in yz plane but rather in xy plane..please tell me how i am wrong?
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:55
add a comment |
$begingroup$
Figures are generally not drawn to scale
$endgroup$
– Shubham Johri
Dec 11 '18 at 16:50
$begingroup$
agreed...but there should at least be some coherence to what is being claimed in textbook answer, because if the bisecting line has to be perpendicular to x-axis it cannot exist in yz plane but rather in xy plane..please tell me how i am wrong?
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:55
$begingroup$
Figures are generally not drawn to scale
$endgroup$
– Shubham Johri
Dec 11 '18 at 16:50
$begingroup$
Figures are generally not drawn to scale
$endgroup$
– Shubham Johri
Dec 11 '18 at 16:50
$begingroup$
agreed...but there should at least be some coherence to what is being claimed in textbook answer, because if the bisecting line has to be perpendicular to x-axis it cannot exist in yz plane but rather in xy plane..please tell me how i am wrong?
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:55
$begingroup$
agreed...but there should at least be some coherence to what is being claimed in textbook answer, because if the bisecting line has to be perpendicular to x-axis it cannot exist in yz plane but rather in xy plane..please tell me how i am wrong?
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:55
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Look at this photo of a book:
The two edges of the book meeting at a corner do not meet at 90 degrees in the photo.
What we learn from this simple example is the projection from 3 dimensions to 2 dimensions generally does not preserve angles.
On the other hand, the $x$-axis is perpendicular to both the $y$ and $z$ axes, and hence (by bi-linearity of dot product, if you like) is perpendicular to any linear combination of them, hence to their bisector.
$endgroup$
$begingroup$
ok by that logic even if the bisecting line exists in xy or xz plane it will still be perpendicular to all the axes from a 3d perspective..like that of a cube
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:59
$begingroup$
@ShaikhSakib: yes, the line perpendicular to the plane is perpendicular to any line in that plane.
$endgroup$
– Vasya
Dec 11 '18 at 17:01
$begingroup$
hmm..that explains it..have accepted your answer @JohnHughes. Thank you.
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 17:03
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Look at this photo of a book:
The two edges of the book meeting at a corner do not meet at 90 degrees in the photo.
What we learn from this simple example is the projection from 3 dimensions to 2 dimensions generally does not preserve angles.
On the other hand, the $x$-axis is perpendicular to both the $y$ and $z$ axes, and hence (by bi-linearity of dot product, if you like) is perpendicular to any linear combination of them, hence to their bisector.
$endgroup$
$begingroup$
ok by that logic even if the bisecting line exists in xy or xz plane it will still be perpendicular to all the axes from a 3d perspective..like that of a cube
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:59
$begingroup$
@ShaikhSakib: yes, the line perpendicular to the plane is perpendicular to any line in that plane.
$endgroup$
– Vasya
Dec 11 '18 at 17:01
$begingroup$
hmm..that explains it..have accepted your answer @JohnHughes. Thank you.
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 17:03
add a comment |
$begingroup$
Look at this photo of a book:
The two edges of the book meeting at a corner do not meet at 90 degrees in the photo.
What we learn from this simple example is the projection from 3 dimensions to 2 dimensions generally does not preserve angles.
On the other hand, the $x$-axis is perpendicular to both the $y$ and $z$ axes, and hence (by bi-linearity of dot product, if you like) is perpendicular to any linear combination of them, hence to their bisector.
$endgroup$
$begingroup$
ok by that logic even if the bisecting line exists in xy or xz plane it will still be perpendicular to all the axes from a 3d perspective..like that of a cube
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:59
$begingroup$
@ShaikhSakib: yes, the line perpendicular to the plane is perpendicular to any line in that plane.
$endgroup$
– Vasya
Dec 11 '18 at 17:01
$begingroup$
hmm..that explains it..have accepted your answer @JohnHughes. Thank you.
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 17:03
add a comment |
$begingroup$
Look at this photo of a book:
The two edges of the book meeting at a corner do not meet at 90 degrees in the photo.
What we learn from this simple example is the projection from 3 dimensions to 2 dimensions generally does not preserve angles.
On the other hand, the $x$-axis is perpendicular to both the $y$ and $z$ axes, and hence (by bi-linearity of dot product, if you like) is perpendicular to any linear combination of them, hence to their bisector.
$endgroup$
Look at this photo of a book:
The two edges of the book meeting at a corner do not meet at 90 degrees in the photo.
What we learn from this simple example is the projection from 3 dimensions to 2 dimensions generally does not preserve angles.
On the other hand, the $x$-axis is perpendicular to both the $y$ and $z$ axes, and hence (by bi-linearity of dot product, if you like) is perpendicular to any linear combination of them, hence to their bisector.
answered Dec 11 '18 at 16:56
John HughesJohn Hughes
64.2k24191
64.2k24191
$begingroup$
ok by that logic even if the bisecting line exists in xy or xz plane it will still be perpendicular to all the axes from a 3d perspective..like that of a cube
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:59
$begingroup$
@ShaikhSakib: yes, the line perpendicular to the plane is perpendicular to any line in that plane.
$endgroup$
– Vasya
Dec 11 '18 at 17:01
$begingroup$
hmm..that explains it..have accepted your answer @JohnHughes. Thank you.
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 17:03
add a comment |
$begingroup$
ok by that logic even if the bisecting line exists in xy or xz plane it will still be perpendicular to all the axes from a 3d perspective..like that of a cube
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:59
$begingroup$
@ShaikhSakib: yes, the line perpendicular to the plane is perpendicular to any line in that plane.
$endgroup$
– Vasya
Dec 11 '18 at 17:01
$begingroup$
hmm..that explains it..have accepted your answer @JohnHughes. Thank you.
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 17:03
$begingroup$
ok by that logic even if the bisecting line exists in xy or xz plane it will still be perpendicular to all the axes from a 3d perspective..like that of a cube
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:59
$begingroup$
ok by that logic even if the bisecting line exists in xy or xz plane it will still be perpendicular to all the axes from a 3d perspective..like that of a cube
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:59
$begingroup$
@ShaikhSakib: yes, the line perpendicular to the plane is perpendicular to any line in that plane.
$endgroup$
– Vasya
Dec 11 '18 at 17:01
$begingroup$
@ShaikhSakib: yes, the line perpendicular to the plane is perpendicular to any line in that plane.
$endgroup$
– Vasya
Dec 11 '18 at 17:01
$begingroup$
hmm..that explains it..have accepted your answer @JohnHughes. Thank you.
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 17:03
$begingroup$
hmm..that explains it..have accepted your answer @JohnHughes. Thank you.
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 17:03
add a comment |
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$begingroup$
Figures are generally not drawn to scale
$endgroup$
– Shubham Johri
Dec 11 '18 at 16:50
$begingroup$
agreed...but there should at least be some coherence to what is being claimed in textbook answer, because if the bisecting line has to be perpendicular to x-axis it cannot exist in yz plane but rather in xy plane..please tell me how i am wrong?
$endgroup$
– Shaikh Sakib
Dec 11 '18 at 16:55