Mixing
About Elements of a Vector Space, and Line Segments between them
$begingroup$
I need complementary explanations for some basic definitions I encountered during introductory Geometry works.
My first question:
Given $V$ any vector space over a field $mathbb{F}$, when we cite any element $P$ of $V$, does it naturally define a vector $OP$? or does it define just a point in the Space?
My second question:
When we speak of line segments, e.g. $t_1 P_1 + t_2 P_2$ with $t_1 + t_2 = 1$ where $t_{i=1,2}$ are scalars, has this line segment any sense of orientation it implies in the Space?
Thanks.
linear-algebra analytic-geometry
asked Jan 27 at 13:38
freehumoristfreehumorist
351214
$endgroup$
|
show 4 more comments
$begingroup$
I need complementary explanations for some basic definitions I encountered during introductory Geometry works.
My first question:
Given $V$ any vector space over a field $mathbb{F}$, when we cite any element $P$ of $V$, does it naturally define a vector $OP$? or does it define just a point in the Space?
My second question:
When we speak of line segments, e.g. $t_1 P_1 + t_2 P_2$ with $t_1 + t_2 = 1$ where $t_{i=1,2}$ are scalars, has this line segment any sense of orientation it implies in the Space?
Thanks.
linear-algebra analytic-geometry
asked Jan 27 at 13:38
freehumoristfreehumorist
351214
$endgroup$
1
$begingroup$
Throw away your knowledge about the point $ O $. If $ P $ is an element of $ V $ then $ P $ is what we call vector. We can add vectors and multiply them by elements of the field.
$endgroup$
– CrabMan
Jan 27 at 13:42
1
$begingroup$
Usually people mean just the set of points when they say line segment.
$endgroup$
– CrabMan
Jan 27 at 13:43
$begingroup$
so that line segments needn't to have any orientation right?
$endgroup$
– freehumorist
Jan 27 at 13:44
1
$begingroup$
For any vectors $ v, u $ the line segment between $ v $ and $ u $ is the same as the line segment between $ u $ and $ v $.
$endgroup$
– CrabMan
Jan 27 at 13:46
1
$begingroup$
You might note that your definition of "line segment" isn't quite right - if $P_1ne P_2$ then what you define is the whole line containing $P_1$ and $P_2$. To get the line segment you need to add the restriction $t_ige0$. (Hence the notion of "line segment" is undefined in a vector space over an arbitrary field, since $t_ige0$ is meaningless.)
$endgroup$
– David C. Ullrich
Jan 27 at 16:35
|
show 4 more comments
$begingroup$
I need complementary explanations for some basic definitions I encountered during introductory Geometry works.
My first question:
Given $V$ any vector space over a field $mathbb{F}$, when we cite any element $P$ of $V$, does it naturally define a vector $OP$? or does it define just a point in the Space?
My second question:
When we speak of line segments, e.g. $t_1 P_1 + t_2 P_2$ with $t_1 + t_2 = 1$ where $t_{i=1,2}$ are scalars, has this line segment any sense of orientation it implies in the Space?
Thanks.
linear-algebra analytic-geometry
asked Jan 27 at 13:38
freehumoristfreehumorist
351214
$endgroup$
I need complementary explanations for some basic definitions I encountered during introductory Geometry works.
My first question:
Given $V$ any vector space over a field $mathbb{F}$, when we cite any element $P$ of $V$, does it naturally define a vector $OP$? or does it define just a point in the Space?
My second question:
When we speak of line segments, e.g. $t_1 P_1 + t_2 P_2$ with $t_1 + t_2 = 1$ where $t_{i=1,2}$ are scalars, has this line segment any sense of orientation it implies in the Space?
Thanks.
linear-algebra analytic-geometry
linear-algebra analytic-geometry
asked Jan 27 at 13:38
freehumoristfreehumorist
351214
asked Jan 27 at 13:38
freehumoristfreehumorist
351214
asked Jan 27 at 13:38
freehumoristfreehumorist
351214
asked Jan 27 at 13:38
freehumoristfreehumorist
351214
asked Jan 27 at 13:38
freehumoristfreehumorist
351214
351214
1
$begingroup$
Throw away your knowledge about the point $ O $. If $ P $ is an element of $ V $ then $ P $ is what we call vector. We can add vectors and multiply them by elements of the field.
$endgroup$
– CrabMan
Jan 27 at 13:42
1
$begingroup$
Usually people mean just the set of points when they say line segment.
$endgroup$
– CrabMan
Jan 27 at 13:43
$begingroup$
so that line segments needn't to have any orientation right?
$endgroup$
– freehumorist
Jan 27 at 13:44
1
$begingroup$
For any vectors $ v, u $ the line segment between $ v $ and $ u $ is the same as the line segment between $ u $ and $ v $.
$endgroup$
– CrabMan
Jan 27 at 13:46
1
$begingroup$
You might note that your definition of "line segment" isn't quite right - if $P_1ne P_2$ then what you define is the whole line containing $P_1$ and $P_2$. To get the line segment you need to add the restriction $t_ige0$. (Hence the notion of "line segment" is undefined in a vector space over an arbitrary field, since $t_ige0$ is meaningless.)
$endgroup$
– David C. Ullrich
Jan 27 at 16:35
|
show 4 more comments
1
$begingroup$
Throw away your knowledge about the point $ O $. If $ P $ is an element of $ V $ then $ P $ is what we call vector. We can add vectors and multiply them by elements of the field.
$endgroup$
– CrabMan
Jan 27 at 13:42
1
$begingroup$
Usually people mean just the set of points when they say line segment.
$endgroup$
– CrabMan
Jan 27 at 13:43
$begingroup$
so that line segments needn't to have any orientation right?
$endgroup$
– freehumorist
Jan 27 at 13:44
1
$begingroup$
For any vectors $ v, u $ the line segment between $ v $ and $ u $ is the same as the line segment between $ u $ and $ v $.
$endgroup$
– CrabMan
Jan 27 at 13:46
1
$begingroup$
You might note that your definition of "line segment" isn't quite right - if $P_1ne P_2$ then what you define is the whole line containing $P_1$ and $P_2$. To get the line segment you need to add the restriction $t_ige0$. (Hence the notion of "line segment" is undefined in a vector space over an arbitrary field, since $t_ige0$ is meaningless.)
$endgroup$
– David C. Ullrich
Jan 27 at 16:35
1
1
$begingroup$
Throw away your knowledge about the point $ O $. If $ P $ is an element of $ V $ then $ P $ is what we call vector. We can add vectors and multiply them by elements of the field.
$endgroup$
– CrabMan
Jan 27 at 13:42
$begingroup$
Throw away your knowledge about the point $ O $. If $ P $ is an element of $ V $ then $ P $ is what we call vector. We can add vectors and multiply them by elements of the field.
$endgroup$
– CrabMan
Jan 27 at 13:42
1
1
$begingroup$
Usually people mean just the set of points when they say line segment.
$endgroup$
– CrabMan
Jan 27 at 13:43
$begingroup$
Usually people mean just the set of points when they say line segment.
$endgroup$
– CrabMan
Jan 27 at 13:43
$begingroup$
so that line segments needn't to have any orientation right?
$endgroup$
– freehumorist
Jan 27 at 13:44
$begingroup$
so that line segments needn't to have any orientation right?
$endgroup$
– freehumorist
Jan 27 at 13:44
1
1
$begingroup$
For any vectors $ v, u $ the line segment between $ v $ and $ u $ is the same as the line segment between $ u $ and $ v $.
$endgroup$
– CrabMan
Jan 27 at 13:46
$begingroup$
For any vectors $ v, u $ the line segment between $ v $ and $ u $ is the same as the line segment between $ u $ and $ v $.
$endgroup$
– CrabMan
Jan 27 at 13:46
1
1
$begingroup$
You might note that your definition of "line segment" isn't quite right - if $P_1ne P_2$ then what you define is the whole line containing $P_1$ and $P_2$. To get the line segment you need to add the restriction $t_ige0$. (Hence the notion of "line segment" is undefined in a vector space over an arbitrary field, since $t_ige0$ is meaningless.)
$endgroup$
– David C. Ullrich
Jan 27 at 16:35
$begingroup$
You might note that your definition of "line segment" isn't quite right - if $P_1ne P_2$ then what you define is the whole line containing $P_1$ and $P_2$. To get the line segment you need to add the restriction $t_ige0$. (Hence the notion of "line segment" is undefined in a vector space over an arbitrary field, since $t_ige0$ is meaningless.)
$endgroup$
– David C. Ullrich
Jan 27 at 16:35
|
show 4 more comments
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1
$begingroup$
Throw away your knowledge about the point $ O $. If $ P $ is an element of $ V $ then $ P $ is what we call vector. We can add vectors and multiply them by elements of the field.
$endgroup$
– CrabMan
Jan 27 at 13:42
1
$begingroup$
Usually people mean just the set of points when they say line segment.
$endgroup$
– CrabMan
Jan 27 at 13:43
$begingroup$
so that line segments needn't to have any orientation right?
$endgroup$
– freehumorist
Jan 27 at 13:44
1
$begingroup$
For any vectors $ v, u $ the line segment between $ v $ and $ u $ is the same as the line segment between $ u $ and $ v $.
$endgroup$
– CrabMan
Jan 27 at 13:46
1
$begingroup$
You might note that your definition of "line segment" isn't quite right - if $P_1ne P_2$ then what you define is the whole line containing $P_1$ and $P_2$. To get the line segment you need to add the restriction $t_ige0$. (Hence the notion of "line segment" is undefined in a vector space over an arbitrary field, since $t_ige0$ is meaningless.)
$endgroup$
– David C. Ullrich
Jan 27 at 16:35