Understanding Vector

[Vastor]

Hey eveyone,

 

until now I can't figured out what is "unit vector" in term of this formula:

 

[math] \hat{r} = \frac{\vec{AB}}{||\vec{AB}||} [/math]

 

especially when referring to the graph, unlike one of the components that it used [math] ||\vec{AB}|| [/math], which referring to the "magnitude/length of a vector".

 

 

another question,

 

10. Given that [math] \vec{OA} = \lambda i[/math] and [math] \vec{OB} = 2i + 3j [/math] and [math] \vec{OC} = 3i - 4j[/math]. Find the value of [math] \lambda [/math] if A, B and C are collinear.

 

(note that i and j are the cartesian coordinates unit vector)

 

per said that "A, B and C are collinear", I plot it like this, assuming the O is the origin.

 

I just not understand what is the *lambda* are really referring too, did I'm missing something here? (btw, the answer is 17/7)

Edited May 30, 2012 by Vastor

[D H]

I just not understand what is the *lambda* are really referring too, did I'm missing something here? (btw, the answer is 17/7)

It's just a parameter. Perhaps it would help if you rewrote it so that [imath]\vec{OA} = x\hat i[/imath] and you solve for x rather than λ. In the end it doesn't matter what you call it.

 

Note that your graph is incorrect. There is no y component in the point A. The coordinates of the point A are (λ, 0).

[imatfaal]

On the first bit of your question

 

[math] \vec{OC} = 3i - 4j [/math] think about what this means. It is the line from O to C - it has a direction and a magnitude (the length of the line) , the unit vector is a vector that goes in the same direction but is only 1 unit in length/magnitude

[Vastor]

It's just a parameter. Perhaps it would help if you rewrote it so that [imath]\vec{OA} = x\hat i[/imath] and you solve for x rather than λ. In the end it doesn't matter what you call it.

 

Note that your graph is incorrect. There is no y component in the point A. The coordinates of the point A are (λ, 0).

Thanks, I took some time to figure out about this because of my presumption told me that the collinearity of the point must be ABC or CBA instead of CAB or others.

 

 

On the first bit of your question

 

[math] \vec{OC} = 3i - 4j [/math] think about what this means. It is the line from O to C - it has a direction and a magnitude (the length of the line) , the unit vector is a vector that goes in the same direction but is only 1 unit in length/magnitude

 

Almost get this, but an example of application in any kind of calculation would help me understand more, I mean [math] \vec{OC} [/math] used to determine the magnitude of something. So, what does the 'unit vector' used for?

 

Anyway, based on my understanding, correct me if I'm wrong, the unit vector for of cartesian coordinate (i, j) is actually the "unit vector" that we are talking about. so basically it used as a scale?