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PLAGUE

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- TL;DR Summary
- If P and R are two points and v is a vector, then when will ##P + tv## and ##R + sv## coincide? Here t and s are parameters that varies over real number.

I was going through this book called "A Course in Mathematics for Students of Physics Volume 1 by Paul Bamberg and Shlomo Sternberg". There in a part they said something like this:

...if we start with a point P and write

##R=P+u##

##Q=P+v##

and

##S=P+(u+v)##

then the four points

##P,Q,S,R##

lie at the four vertices of a parallelogram... The proof of this fact goes as follows. For any vector

##v=(a,b)##

and any real number t defines their product tv by

##tv=(ta,tb)##

if P is any point the set

##l=P+tv##

(as t varies over real number), is a straight line passing through P. If R is some other point, then the line

##m=R+sv##

(as s varies over real number) and l will intersect, i.e., have some point in common, if and only if there are some

##s_1##

and

##t_1##

such that,

##R+s_1v=P+t_1v##

which means that

##R=P+(t_1−s_1)v##

and hence, for every s, that

##R+sv=P+(s+t_1−s_1)v.##

This means that the lines m and l coincide. In other words, either the lines l and m coincide or they do not intersect, i.e., either they are the same or they are parallel...

Now what I don't understand is the last sentence, why m and l coincide? How can they say m and l coincide from the equation,

##R+sv=P+(s+t_1−s_1)v##

?

...if we start with a point P and write

##R=P+u##

##Q=P+v##

and

##S=P+(u+v)##

then the four points

##P,Q,S,R##

lie at the four vertices of a parallelogram... The proof of this fact goes as follows. For any vector

##v=(a,b)##

and any real number t defines their product tv by

##tv=(ta,tb)##

if P is any point the set

##l=P+tv##

(as t varies over real number), is a straight line passing through P. If R is some other point, then the line

##m=R+sv##

(as s varies over real number) and l will intersect, i.e., have some point in common, if and only if there are some

##s_1##

and

##t_1##

such that,

##R+s_1v=P+t_1v##

which means that

##R=P+(t_1−s_1)v##

and hence, for every s, that

##R+sv=P+(s+t_1−s_1)v.##

This means that the lines m and l coincide. In other words, either the lines l and m coincide or they do not intersect, i.e., either they are the same or they are parallel...

Now what I don't understand is the last sentence, why m and l coincide? How can they say m and l coincide from the equation,

##R+sv=P+(s+t_1−s_1)v##

?

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