Hi http://forum.allaboutcircuits.com/attachment.php?attachmentid=34734&stc=1&d=1317507259 But I have been told that we also get the same result even when the two vectors are collinear. Then, how do we know that if they are parallel or collinear? Two parallel straight lines have different y-axis intercepts if they are not collinear. What do you say? Thank you. Regards PG PS: There is some problem with Latex compiler of the forums because I checked the code on some other website and it compiled there correctly but here I got weird compiled result.
Vectors are not the same thing as lines. In order for vectors to be compared, they have to be in the same vector space, and if they are to be compared by components, the vectors must use the same coordinate reference frame. So, what do vectors possess? They have magnitude and direction: that's it. They don't have y-intercepts, or if you want to say they do, the intercept is always zero. Hence all parallel vectors are also co-linear, in the sense you are thinking. They can only differ in magnitude. We sometimes get into the case of vector fields, where each point has a vector associated with it. Strictly speaking, every point has it's own separate vector space, and you really shouldn't try to compare vectors like this. You would instead parallel transport the vectors, until they have the same origin, before you compare them. This may seem confusing at first, but if it this explanation bothers you, then you can take the answer to be that vectors which use a different origin can be parallel and vectors that use the same origin can be colinear. On the issue of the latex compiler, can you post the code you tried to use?
The definition you gave states that, in the basis given, one vector is a multiple of another. As steveb said, vectors have two characteristics: a magnitude and a direction. If they're multiples, then they're pointing in the same direction and are parallel. A good habit to get into is to think of a vector as an arrow in the plane or space (or higher dimensional spaces) and ignore any coordinate systems when you can. The reason is that then you focus on the essence of the behavior. There's a deep reason for this that'll you'll learn later in your studies; it's captured in the concept that the fundamental laws of nature should be covariant, which means the form of the equations doesn't depend on the coordinate system chosen for the problem. This can also be related to the symmetries of the system (those transformations that leave the system unchanged) and leads to Noether's Theorem, one of the most beautiful theorems I've ever studied. The usual test for two vectors being parallel in basic math and science is that their cross product should be zero. The cross product is an ugly construction, since it only exists in three dimensions. To whet your appetite, a much better formulation was discovered by mathematicians in the 19th century such as Clifford and Grassman. This led to the notion of a geometric product in a Clifford algebra, which is an elegant tool, as it a) can be defined in spaces of any dimensions and b) in encompasses and generalizes complex numbers and Hamilton's quaternions. The outer product, which is part of the geometric product, is always zero for linearly dependent vectors.
Thanks a lot, Steve, someonesdad. I might have some follow-on questions which I would ask later. @Steve:- The code is given below. Before compiling it I had surrounded it with
Code ( (Unknown Language)): $ Two\ vectors\ \vec{a}\ and\ \vec{b}\ are\ parallel\ if\ \frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}=\lambda ,\ where\ \vec{a}=a_{1}i+a_{2}j+a_{3}k\;\;\vec{b}=b_{1}i+b_{2}j+b_{3}k $ You can also get rid of the italic words as follows. Code ( (Unknown Language)): $ {\rm Two\ vectors\ }\vec{a}\ {\rm and}\ \vec{b}\ {\rm are\ parallel\ if\ }\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}=\lambda ,\ {\rm where\ }\vec{a}=a_{1}i+a_{2}j+a_{3}k\ {\rm and}\ \vec{b}=b_{1}i+b_{2}j+b_{3}k $ " alt=" Yes, I agree. I can get this working in my laTeX compiler at home, but it doesn't work here. It seems that some of the control sequences are not implemented. You can use alternate commands. For example a \; can be used to make spaces and a backslash followed by a space also works. Also the \vec command will put an arrow on top of the variable. I implemented your code as follows, and I've attached a TeX command cheatsheet which seems to have commands that work, usually. You'll also notice that this is actually quicker to type out. Code ( (Unknown Language)): $ Two\ vectors\ \vec{a}\ and\ \vec{b}\ are\ parallel\ if\ \frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}=\lambda ,\ where\ \vec{a}=a_{1}i+a_{2}j+a_{3}k\;\;\vec{b}=b_{1}i+b_{2}j+b_{3}k $ You can also get rid of the italic words as follows. Code ( (Unknown Language)): $ {\rm Two\ vectors\ }\vec{a}\ {\rm and}\ \vec{b}\ {\rm are\ parallel\ if\ }\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}=\lambda ,\ {\rm where\ }\vec{a}=a_{1}i+a_{2}j+a_{3}k\ {\rm and}\ \vec{b}=b_{1}i+b_{2}j+b_{3}k $ " />
Thank you very much, Steve. You have been helping with so many queries! I'm much obliged. Perhaps, some moderator should take Latex related posts out of this thread and make a Latex specific thread out of them. I have very little knowledge how to write Latex code. As I mentioned in previous post I use Scientific Workplace program which offers me with all the math related symbols which I can simply click on. I can use it either in Text mode or Math mode. The math mode didn't let me insert spaces until yesterday when I came around this and used tab button instead for spaces. If I used both modes to create a Latex text, then it doesn't compile. When I use both Text and Math modes I get this and I have not been able to compile it anywhere on any of the forums I use (also check the attached screenshot to see how it looks in the Scientific Workplace): 038<p type="texpara" tag="Body Math" >Two\hspace{0.06in}vectors$\hspace{0.06in}\overrightarrow{a}\hspace{0.06in}$and$\hspace{0.06in}\overrightarrow{b}\hspace{0.06in}$are\hspace{0.06in}parallel\hspace{0.06in}if$\hspace{0.06in}\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}=\lambda ,\hspace{0.06in}$where$\hspace{0.06in}\overrightarrow{a}=a_{1}i+a_{2}j+a_{3}k\hspace{0.06in}\overrightarrow{b}=b_{1}i+b_{2}j+b_{3}k$ 038<p type="texpara" tag="Body Math" >$\bigskip $Indefinite integral is written as: $\int f(x)dx$ I was fortunate yesterday as I told you that I figured out a way to insert spaces and get a Latex code which at least works on some forums. I needed to divid the code into parts in order to compile it on some other forum because if I didn't I would get this error: Code ( (Unknown Language)): [LaTeX ERROR: Image too big 1063x35, max 1000x600] See, if you can help me with this. And someone should fix the Latex compiler here because many a time you need to use equations. Best regards PG PS: I divided the code this way (I have inserted the dots myself): [laTEX.]$Two\hspace{0.06in}vectors\hspace{0.06in}\overrightarrow{a}\hspace{0.06in}and\hspace{0.06in}\overrightarrow{b}\hspace{0.06in}are\hspace{0.06in}parallel\hspace{0.06in}if\hspace{0.06in}\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}=\lambda ,\hspace{0.06in}[/laTEX][laTEX]where\hspace{0.06in}\overrightarrow{a}=a_{1}i+a_{2}j+a_{3}k\hspace{0.06in}\overrightarrow{b}=b_{1}i+b_{2}j+b_{3}k$[/laTEX.]
For now take a look at those two links that I have found very useful: http://en.wikibooks.org/wiki/LaTeX/Mathematics#Symbols http://amath.colorado.edu/documentation/LaTeX/Symbols.pdf Not all of their functions work, especially the format options. In the near future I plan to make a LaTeX how to for the site.