mameko
  • mameko
Hi, I have some questions regarding to Pset 8 4A-3 b). I don't know how to get the answer. >_<... My approach was first find some points ((-1, 1), (1,1), etc ) and drew them on a graph. Then try to figure out what F looks like by looking at the direction of every vector and the magnitude. Could you tell me what is your way of solving this kind of question ? Thanks in advance.
OCW Scholar - Multivariable Calculus
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schrodinger
  • schrodinger
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phi
  • phi
Are you asking about "Write down an explicit expression for each of the following fields:" b) The vector at (x,y) is directed radially in towards the origin, with magnitude r^2 ? First, we have to understand what they are saying. "radially inward towards the origin" means a vector at location (x,y), pointing directly at the origin. Ignoring the magnitude, it should look like this |dw:1447286273132:dw|
phi
  • phi
Next, if we are given two points, say A and B the vector from A to B is found by B-A In this problem, we know the "tail" of each vector is at the (arbitrary) point (x,y) and the head is (0,0) thus the vector pointing in the direction from (x,y,) to (0,0) is given by (0,0)- (x,y) = -(x,y) (or (-x,-y) but it seems nicer to keep the scalar -1 outside) that gives the correct direction. It is usually nice to normalize a direction vector to unit length (and then scale it to the correct length) based in the info, we can assume r represents the distance of point (x,y) from the origin i.e. the length of the vector from (x,y) to the origin has length r (we are assuming they mean r = sqr(x^2 + y^2) though they don't explicitly say this) but to continue, the unit length vector pointing toward the origin will be \[ - \frac{(x,y)}{r} \] Finally, we are told the required vector has length r^2, so scale the unit length vector by this amount: \[ - \frac{(x,y)}{r} \cdot r^2 = -r (x,y) \] or using the i, j notation (which means the same thing as the (x,y) notation) \[ -r (x\ \textbf{i}+y\ \textbf{j} ) \]
mameko
  • mameko
Thank you very much for replying. I didn't normalize the vector when I was working on the problem, so I got -r^2(xi + yj). I am wondering how did you know that we need to normalize the vector ?

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phi
  • phi
**how did you know that we need to normalize the vector ? ** I tried to explain that: It is usually nice to normalize a direction vector to unit length (and then scale it to the correct length) For example, say it was the same problem but the magnitude was given as 2 instead of r^2 Then the idea is you first make a "direction vector" which means a vector that points in the correct direction by has *unit length* then we multiply that unit length vector by the magnitude we want it to have. So, to get a vector of length 2, we would have -2 (x,y)/r this has the vector pointing in the direction -(x,y) then normalized to unit length: -(x,y)/r then scaled by the magnitude we want : -2(x,y)/r
mameko
  • mameko
Looks like normalizing the vector is the right thing to do. I didn't know that before. Thank you very much. I'll keep that in mind.

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