How many Scalers can be built using three different unit vectors? [closed]
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I have three unit vectors in a problem:
$hatt= (cos(t),0,sin(t)),$
$ hatm= (0,0,1),$
$hatn= (sin(th),0,-cos(th)).$
I know the solution for the problem is:
$(-sin(2t)+ 5 sin(2t-4th)+2 sin(2 th))$
I want to write the answer in a form in which only used variables are unit vectors instead of anges. Could anyone help me?
vector-spaces vectors inner-product-space products
edited Aug 16 at 19:18
Sil
5,14121443
asked Aug 12 at 7:26
Holger Mate
236
closed as unclear what you're asking by Arnaud D., John Ma, José Carlos Santos, Cesareo, Leucippus Aug 17 at 0:22
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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up vote
-1
down vote
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I have three unit vectors in a problem:
$hatt= (cos(t),0,sin(t)),$
$ hatm= (0,0,1),$
$hatn= (sin(th),0,-cos(th)).$
I know the solution for the problem is:
$(-sin(2t)+ 5 sin(2t-4th)+2 sin(2 th))$
I want to write the answer in a form in which only used variables are unit vectors instead of anges. Could anyone help me?
vector-spaces vectors inner-product-space products
edited Aug 16 at 19:18
Sil
5,14121443
asked Aug 12 at 7:26
Holger Mate
236
closed as unclear what you're asking by Arnaud D., John Ma, José Carlos Santos, Cesareo, Leucippus Aug 17 at 0:22
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
Thank you for your comment. I edited my question to make it more clear. Is it ok now? @mr_e_man
– Holger Mate
Aug 12 at 7:47
The middle dimension is never used; all three vectors are in a plane. And you still haven't really given a problem to solve.
– mr_e_man
Aug 12 at 7:51
I tried to make the question more clear again. @mr_e_man
– Holger Mate
Aug 12 at 7:52
Is it oke now? @mr_e_man
– Holger Mate
Aug 12 at 7:53
I guess it makes sense now. Are you sure all the vectors have $y=0$?
– mr_e_man
Aug 12 at 7:54
 |Â
show 1 more comment
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I have three unit vectors in a problem:
$hatt= (cos(t),0,sin(t)),$
$ hatm= (0,0,1),$
$hatn= (sin(th),0,-cos(th)).$
I know the solution for the problem is:
$(-sin(2t)+ 5 sin(2t-4th)+2 sin(2 th))$
I want to write the answer in a form in which only used variables are unit vectors instead of anges. Could anyone help me?
vector-spaces vectors inner-product-space products
edited Aug 16 at 19:18
Sil
5,14121443
asked Aug 12 at 7:26
Holger Mate
236
I have three unit vectors in a problem:
$hatt= (cos(t),0,sin(t)),$
$ hatm= (0,0,1),$
$hatn= (sin(th),0,-cos(th)).$
I know the solution for the problem is:
$(-sin(2t)+ 5 sin(2t-4th)+2 sin(2 th))$
I want to write the answer in a form in which only used variables are unit vectors instead of anges. Could anyone help me?
vector-spaces vectors inner-product-space products
edited Aug 16 at 19:18
Sil
5,14121443
asked Aug 12 at 7:26
Holger Mate
236
edited Aug 16 at 19:18
Sil
5,14121443
edited Aug 16 at 19:18
Sil
5,14121443
edited Aug 16 at 19:18
Sil
5,14121443
5,14121443
asked Aug 12 at 7:26
Holger Mate
236
asked Aug 12 at 7:26
Holger Mate
236
asked Aug 12 at 7:26
Holger Mate
236
236
closed as unclear what you're asking by Arnaud D., John Ma, José Carlos Santos, Cesareo, Leucippus Aug 17 at 0:22
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Arnaud D., John Ma, José Carlos Santos, Cesareo, Leucippus Aug 17 at 0:22
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
Thank you for your comment. I edited my question to make it more clear. Is it ok now? @mr_e_man
– Holger Mate
Aug 12 at 7:47
The middle dimension is never used; all three vectors are in a plane. And you still haven't really given a problem to solve.
– mr_e_man
Aug 12 at 7:51
I tried to make the question more clear again. @mr_e_man
– Holger Mate
Aug 12 at 7:52
Is it oke now? @mr_e_man
– Holger Mate
Aug 12 at 7:53
I guess it makes sense now. Are you sure all the vectors have $y=0$?
– mr_e_man
Aug 12 at 7:54
 |Â
show 1 more comment
Thank you for your comment. I edited my question to make it more clear. Is it ok now? @mr_e_man
– Holger Mate
Aug 12 at 7:47
The middle dimension is never used; all three vectors are in a plane. And you still haven't really given a problem to solve.
– mr_e_man
Aug 12 at 7:51
I tried to make the question more clear again. @mr_e_man
– Holger Mate
Aug 12 at 7:52
Is it oke now? @mr_e_man
– Holger Mate
Aug 12 at 7:53
I guess it makes sense now. Are you sure all the vectors have $y=0$?
– mr_e_man
Aug 12 at 7:54
Thank you for your comment. I edited my question to make it more clear. Is it ok now? @mr_e_man
– Holger Mate
Aug 12 at 7:47
Thank you for your comment. I edited my question to make it more clear. Is it ok now? @mr_e_man
– Holger Mate
Aug 12 at 7:47
The middle dimension is never used; all three vectors are in a plane. And you still haven't really given a problem to solve.
– mr_e_man
Aug 12 at 7:51
The middle dimension is never used; all three vectors are in a plane. And you still haven't really given a problem to solve.
– mr_e_man
Aug 12 at 7:51
I tried to make the question more clear again. @mr_e_man
– Holger Mate
Aug 12 at 7:52
I tried to make the question more clear again. @mr_e_man
– Holger Mate
Aug 12 at 7:52
Is it oke now? @mr_e_man
– Holger Mate
Aug 12 at 7:53
Is it oke now? @mr_e_man
– Holger Mate
Aug 12 at 7:53
I guess it makes sense now. Are you sure all the vectors have $y=0$?
– mr_e_man
Aug 12 at 7:54
I guess it makes sense now. Are you sure all the vectors have $y=0$?
– mr_e_man
Aug 12 at 7:54
 |Â
show 1 more comment
1 Answer
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Your solution can be expanded with the trig identities
$$sin(2theta) = 2sin(theta)cos(theta)$$
$$cos(2theta) = cos^2(theta)-sin^2(theta)$$
$$sin(theta+phi) = sin(theta)cos(phi)+cos(theta)sin(phi)$$
$$cos(theta+phi) = cos(theta)cos(phi)-sin(theta)sin(phi)$$
to be expressed in terms of $sin(t)$, $cos(t)$, $sin(th)$, and $cos(th)$.
We also have
$$hat tcdothat m = sin(t)$$
$$hat ncdothat m = -cos(th)$$
$$hat tcdothat n = sin(th-t)$$
but this doesn't give us a way to express $cos(t)$ or $sin(th)$ with dot products. (We know that $cos(t) = pmsqrt1-sin^2(t)$ $= pmsqrt1-(hat tcdothat m)^2$, but cosine's sign is not determined by the sine.) So I think you must use the components
$$t_x = hat tcdot(1,0,0) = cos(t)$$
$$n_x = hat ncdot(1,0,0) = sin(th)$$
answered Aug 12 at 8:17
mr_e_man
854420
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Your solution can be expanded with the trig identities
$$sin(2theta) = 2sin(theta)cos(theta)$$
$$cos(2theta) = cos^2(theta)-sin^2(theta)$$
$$sin(theta+phi) = sin(theta)cos(phi)+cos(theta)sin(phi)$$
$$cos(theta+phi) = cos(theta)cos(phi)-sin(theta)sin(phi)$$
to be expressed in terms of $sin(t)$, $cos(t)$, $sin(th)$, and $cos(th)$.
We also have
$$hat tcdothat m = sin(t)$$
$$hat ncdothat m = -cos(th)$$
$$hat tcdothat n = sin(th-t)$$
but this doesn't give us a way to express $cos(t)$ or $sin(th)$ with dot products. (We know that $cos(t) = pmsqrt1-sin^2(t)$ $= pmsqrt1-(hat tcdothat m)^2$, but cosine's sign is not determined by the sine.) So I think you must use the components
$$t_x = hat tcdot(1,0,0) = cos(t)$$
$$n_x = hat ncdot(1,0,0) = sin(th)$$
answered Aug 12 at 8:17
mr_e_man
854420
add a comment |Â
up vote
0
down vote
accepted
Your solution can be expanded with the trig identities
$$sin(2theta) = 2sin(theta)cos(theta)$$
$$cos(2theta) = cos^2(theta)-sin^2(theta)$$
$$sin(theta+phi) = sin(theta)cos(phi)+cos(theta)sin(phi)$$
$$cos(theta+phi) = cos(theta)cos(phi)-sin(theta)sin(phi)$$
to be expressed in terms of $sin(t)$, $cos(t)$, $sin(th)$, and $cos(th)$.
We also have
$$hat tcdothat m = sin(t)$$
$$hat ncdothat m = -cos(th)$$
$$hat tcdothat n = sin(th-t)$$
but this doesn't give us a way to express $cos(t)$ or $sin(th)$ with dot products. (We know that $cos(t) = pmsqrt1-sin^2(t)$ $= pmsqrt1-(hat tcdothat m)^2$, but cosine's sign is not determined by the sine.) So I think you must use the components
$$t_x = hat tcdot(1,0,0) = cos(t)$$
$$n_x = hat ncdot(1,0,0) = sin(th)$$
answered Aug 12 at 8:17
mr_e_man
854420
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Your solution can be expanded with the trig identities
$$sin(2theta) = 2sin(theta)cos(theta)$$
$$cos(2theta) = cos^2(theta)-sin^2(theta)$$
$$sin(theta+phi) = sin(theta)cos(phi)+cos(theta)sin(phi)$$
$$cos(theta+phi) = cos(theta)cos(phi)-sin(theta)sin(phi)$$
to be expressed in terms of $sin(t)$, $cos(t)$, $sin(th)$, and $cos(th)$.
We also have
$$hat tcdothat m = sin(t)$$
$$hat ncdothat m = -cos(th)$$
$$hat tcdothat n = sin(th-t)$$
but this doesn't give us a way to express $cos(t)$ or $sin(th)$ with dot products. (We know that $cos(t) = pmsqrt1-sin^2(t)$ $= pmsqrt1-(hat tcdothat m)^2$, but cosine's sign is not determined by the sine.) So I think you must use the components
$$t_x = hat tcdot(1,0,0) = cos(t)$$
$$n_x = hat ncdot(1,0,0) = sin(th)$$
answered Aug 12 at 8:17
mr_e_man
854420
Your solution can be expanded with the trig identities
$$sin(2theta) = 2sin(theta)cos(theta)$$
$$cos(2theta) = cos^2(theta)-sin^2(theta)$$
$$sin(theta+phi) = sin(theta)cos(phi)+cos(theta)sin(phi)$$
$$cos(theta+phi) = cos(theta)cos(phi)-sin(theta)sin(phi)$$
to be expressed in terms of $sin(t)$, $cos(t)$, $sin(th)$, and $cos(th)$.
We also have
$$hat tcdothat m = sin(t)$$
$$hat ncdothat m = -cos(th)$$
$$hat tcdothat n = sin(th-t)$$
but this doesn't give us a way to express $cos(t)$ or $sin(th)$ with dot products. (We know that $cos(t) = pmsqrt1-sin^2(t)$ $= pmsqrt1-(hat tcdothat m)^2$, but cosine's sign is not determined by the sine.) So I think you must use the components
$$t_x = hat tcdot(1,0,0) = cos(t)$$
$$n_x = hat ncdot(1,0,0) = sin(th)$$
answered Aug 12 at 8:17
mr_e_man
854420
answered Aug 12 at 8:17
mr_e_man
854420
answered Aug 12 at 8:17
mr_e_man
854420
answered Aug 12 at 8:17
mr_e_man
854420
854420
add a comment |Â
add a comment |Â
Thank you for your comment. I edited my question to make it more clear. Is it ok now? @mr_e_man
– Holger Mate
Aug 12 at 7:47
The middle dimension is never used; all three vectors are in a plane. And you still haven't really given a problem to solve.
– mr_e_man
Aug 12 at 7:51
I tried to make the question more clear again. @mr_e_man
– Holger Mate
Aug 12 at 7:52
Is it oke now? @mr_e_man
– Holger Mate
Aug 12 at 7:53
I guess it makes sense now. Are you sure all the vectors have $y=0$?
– mr_e_man
Aug 12 at 7:54