Question

Hi, I have a question about Multivariable Pset 8 Part 2 Problem 1 a). In the answer, point P goes from below the x + y = 1 to that line. so we have F = (1 - x- y) j . When I was working on the problem, I put point P above that line. So I got F = (y - 1 + x) j instead. Why should we choose P below the line instead of above the line ? Thanks in advance.

Answer 1

the field will have a direction <0,1> (or 0 i + 1 j if you prefer)
but scaled by a magnitude (i.e. the difference between the y value of the point in the field and the point on the line), and of course with the correct sense (pointing up or down)

if we compute the magnitude at point (Px,Py) to the line at (Px, P_line)
as P_line - Py this will give a negative number if the point is above the line.
that is, we have
\[ -|P_{line}-P_y|<0,1> = |P_{line}-P_y|<0,-1>
\]
and we see the vector has the correct direction, length and sense (i. e. down)

if we have the point below the line, we will get a positive magnitude and the field vector will be
\[ |P_{line}-P_y|<0,1> \]
which is also correct i.e. we are now point up.


when you **, I put point P above that line.***
you are implicitly saying the direction vector is <0, -1> i..e point down
and then you got (y - 1 + x) j
It looks like you lost the -1.
For example, say you are at (1,1)
the line is at (1,0), and your formula gives (1-1+1)j = +1j
i.e. <0,1> which points up and you want to point down.

if you were to say
(y-1+x) is the magnitude and sense that you then scale <0,-1>
you would get
<0, 1-x-y>
or
(1-x-y) j
and that is the correct vector.

Answer 2

Yes, right. That make sense now. I only calculated the magnitude but didn't pay attention to the direction of that vector. Thank you very much for your help.