Finding sign of an angle without calculating the angle itself
Say we have an angle between vectors $P_1$ and $P_2$ in $mathbb R^2$ whose vertex, $O$, is at the origin. I would like to know the sign of the smallest signed angle between $P_1$ and $P_2$ such that $overrightarrow{O{P_1}}$ is rotated to become overlaid on $overrightarrow{O{P_2}}$. It is too computationally expensive for me to use trigonometric functions to calculate the angle itself and then find the sign. Is there a good way to do this?
linear-algebra geometry rotations angle
edited Nov 26 '18 at 1:21
KReiser
9,31211435
asked Nov 25 '18 at 22:54
rbjacob
223
add a comment |
Say we have an angle between vectors $P_1$ and $P_2$ in $mathbb R^2$ whose vertex, $O$, is at the origin. I would like to know the sign of the smallest signed angle between $P_1$ and $P_2$ such that $overrightarrow{O{P_1}}$ is rotated to become overlaid on $overrightarrow{O{P_2}}$. It is too computationally expensive for me to use trigonometric functions to calculate the angle itself and then find the sign. Is there a good way to do this?
linear-algebra geometry rotations angle
edited Nov 26 '18 at 1:21
KReiser
9,31211435
asked Nov 25 '18 at 22:54
rbjacob
223
Do you mean find the sine or the sign?
– Seth
Nov 25 '18 at 23:04
The sign as in positive or negative
– rbjacob
Nov 25 '18 at 23:06
add a comment |
Say we have an angle between vectors $P_1$ and $P_2$ in $mathbb R^2$ whose vertex, $O$, is at the origin. I would like to know the sign of the smallest signed angle between $P_1$ and $P_2$ such that $overrightarrow{O{P_1}}$ is rotated to become overlaid on $overrightarrow{O{P_2}}$. It is too computationally expensive for me to use trigonometric functions to calculate the angle itself and then find the sign. Is there a good way to do this?
linear-algebra geometry rotations angle
edited Nov 26 '18 at 1:21
KReiser
9,31211435
asked Nov 25 '18 at 22:54
rbjacob
223
Say we have an angle between vectors $P_1$ and $P_2$ in $mathbb R^2$ whose vertex, $O$, is at the origin. I would like to know the sign of the smallest signed angle between $P_1$ and $P_2$ such that $overrightarrow{O{P_1}}$ is rotated to become overlaid on $overrightarrow{O{P_2}}$. It is too computationally expensive for me to use trigonometric functions to calculate the angle itself and then find the sign. Is there a good way to do this?
linear-algebra geometry rotations angle
linear-algebra geometry rotations angle
edited Nov 26 '18 at 1:21
KReiser
9,31211435
asked Nov 25 '18 at 22:54
rbjacob
223
edited Nov 26 '18 at 1:21
KReiser
9,31211435
asked Nov 25 '18 at 22:54
rbjacob
223
edited Nov 26 '18 at 1:21
KReiser
9,31211435
edited Nov 26 '18 at 1:21
KReiser
9,31211435
edited Nov 26 '18 at 1:21
KReiser
9,31211435
9,31211435
asked Nov 25 '18 at 22:54
rbjacob
223
asked Nov 25 '18 at 22:54
rbjacob
223
asked Nov 25 '18 at 22:54
rbjacob
223
223
Do you mean find the sine or the sign?
– Seth
Nov 25 '18 at 23:04
The sign as in positive or negative
– rbjacob
Nov 25 '18 at 23:06
add a comment |
Do you mean find the sine or the sign?
– Seth
Nov 25 '18 at 23:04
The sign as in positive or negative
– rbjacob
Nov 25 '18 at 23:06
Do you mean find the sine or the sign?
– Seth
Nov 25 '18 at 23:04
Do you mean find the sine or the sign?
– Seth
Nov 25 '18 at 23:04
The sign as in positive or negative
– rbjacob
Nov 25 '18 at 23:06
The sign as in positive or negative
– rbjacob
Nov 25 '18 at 23:06
add a comment |
1 Answer
1
active
oldest
votes
Check the sign of the coefficient of $hat{k}$ of the cross product of both vectors.
For example, let $P_1=(5,0)$, $P_2=(-3,6)$ , $overrightarrow{OP_1}times overrightarrow{OP_2}=30hat{k}$, so it's a counterclockwise smallest turn. Switching both vectors will result in a negative coefficient for $hat{k}$, indicating a clockwise smallest turn.
Basically for $P_1(a,b), P_2(c,d)$, and turning from $overrightarrow{OP_1}$ to $overrightarrow{OP_2}$, you test sign of $ad-bc$ for the direction of smallest turn (positive for counterclockwise). In the case the result is $0$, either way is the same.
answered Nov 25 '18 at 23:11
Lance
61229
Thanks, perfect
– rbjacob
Nov 25 '18 at 23:19
1
In other words, examine $det[P_1,P_2]$.
– amd
Nov 25 '18 at 23:56
@amd Thanks for the insight!
– Lance
Nov 26 '18 at 0:38
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Check the sign of the coefficient of $hat{k}$ of the cross product of both vectors.
For example, let $P_1=(5,0)$, $P_2=(-3,6)$ , $overrightarrow{OP_1}times overrightarrow{OP_2}=30hat{k}$, so it's a counterclockwise smallest turn. Switching both vectors will result in a negative coefficient for $hat{k}$, indicating a clockwise smallest turn.
Basically for $P_1(a,b), P_2(c,d)$, and turning from $overrightarrow{OP_1}$ to $overrightarrow{OP_2}$, you test sign of $ad-bc$ for the direction of smallest turn (positive for counterclockwise). In the case the result is $0$, either way is the same.
answered Nov 25 '18 at 23:11
Lance
61229
Thanks, perfect
– rbjacob
Nov 25 '18 at 23:19
1
In other words, examine $det[P_1,P_2]$.
– amd
Nov 25 '18 at 23:56
@amd Thanks for the insight!
– Lance
Nov 26 '18 at 0:38
add a comment |
Check the sign of the coefficient of $hat{k}$ of the cross product of both vectors.
For example, let $P_1=(5,0)$, $P_2=(-3,6)$ , $overrightarrow{OP_1}times overrightarrow{OP_2}=30hat{k}$, so it's a counterclockwise smallest turn. Switching both vectors will result in a negative coefficient for $hat{k}$, indicating a clockwise smallest turn.
Basically for $P_1(a,b), P_2(c,d)$, and turning from $overrightarrow{OP_1}$ to $overrightarrow{OP_2}$, you test sign of $ad-bc$ for the direction of smallest turn (positive for counterclockwise). In the case the result is $0$, either way is the same.
answered Nov 25 '18 at 23:11
Lance
61229
Thanks, perfect
– rbjacob
Nov 25 '18 at 23:19
1
In other words, examine $det[P_1,P_2]$.
– amd
Nov 25 '18 at 23:56
@amd Thanks for the insight!
– Lance
Nov 26 '18 at 0:38
add a comment |
Check the sign of the coefficient of $hat{k}$ of the cross product of both vectors.
For example, let $P_1=(5,0)$, $P_2=(-3,6)$ , $overrightarrow{OP_1}times overrightarrow{OP_2}=30hat{k}$, so it's a counterclockwise smallest turn. Switching both vectors will result in a negative coefficient for $hat{k}$, indicating a clockwise smallest turn.
Basically for $P_1(a,b), P_2(c,d)$, and turning from $overrightarrow{OP_1}$ to $overrightarrow{OP_2}$, you test sign of $ad-bc$ for the direction of smallest turn (positive for counterclockwise). In the case the result is $0$, either way is the same.
answered Nov 25 '18 at 23:11
Lance
61229
Check the sign of the coefficient of $hat{k}$ of the cross product of both vectors.
For example, let $P_1=(5,0)$, $P_2=(-3,6)$ , $overrightarrow{OP_1}times overrightarrow{OP_2}=30hat{k}$, so it's a counterclockwise smallest turn. Switching both vectors will result in a negative coefficient for $hat{k}$, indicating a clockwise smallest turn.
Basically for $P_1(a,b), P_2(c,d)$, and turning from $overrightarrow{OP_1}$ to $overrightarrow{OP_2}$, you test sign of $ad-bc$ for the direction of smallest turn (positive for counterclockwise). In the case the result is $0$, either way is the same.
answered Nov 25 '18 at 23:11
Lance
61229
edited Nov 25 '18 at 23:16
answered Nov 25 '18 at 23:11
Lance
61229
answered Nov 25 '18 at 23:11
Lance
61229
answered Nov 25 '18 at 23:11
Lance
61229
61229
Thanks, perfect
– rbjacob
Nov 25 '18 at 23:19
1
In other words, examine $det[P_1,P_2]$.
– amd
Nov 25 '18 at 23:56
@amd Thanks for the insight!
– Lance
Nov 26 '18 at 0:38
add a comment |
Thanks, perfect
– rbjacob
Nov 25 '18 at 23:19
1
In other words, examine $det[P_1,P_2]$.
– amd
Nov 25 '18 at 23:56
@amd Thanks for the insight!
– Lance
Nov 26 '18 at 0:38
Thanks, perfect
– rbjacob
Nov 25 '18 at 23:19
Thanks, perfect
– rbjacob
Nov 25 '18 at 23:19
1
1
In other words, examine $det[P_1,P_2]$.
– amd
Nov 25 '18 at 23:56
In other words, examine $det[P_1,P_2]$.
– amd
Nov 25 '18 at 23:56
@amd Thanks for the insight!
– Lance
Nov 26 '18 at 0:38
@amd Thanks for the insight!
– Lance
Nov 26 '18 at 0:38
add a comment |
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Do you mean find the sine or the sign?
– Seth
Nov 25 '18 at 23:04
The sign as in positive or negative
– rbjacob
Nov 25 '18 at 23:06