- #1
Saladsamurai
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Okay. My reason for posting this is that I need help actually formulating the 'math part' of it. I can get the right answer by 'inspection.' And from the way the book is written, I believe that is how the authors expect you to find it. But for self gratifying reasons, I wish to generalize the answer a little bit:
Write the formula for a vector function in 2 dimensions whose direction makes an
angle of 45 degrees with the x-axis and whose magnitude at any point (x,y) is (x+y)2.
Now just looking at it, we can say that to insure that the vector at any point (x,y) is at 45 degrees to the x-axis, we can let the funciton equal something to the effect of:
[tex]\mathbf{F}(x,y) = k(\bold{i} + \bold{j})[/tex]
Now k(x,y) is the 'scalar portion' of F and must have the effect that its product with the magnitude of the
vector portion of F, namely v=(i + j), must equal (x+y)2.
Now since the magnitude of v=(i + j) is [itex]\sqrt{2}[/itex], k(x,y) must have [itex]\sqrt{2}[/itex] in its denominator.
This will make 'unit vector' in the v direction or
[tex]\bold{u}_v = \frac{\bold{v}}{\sqrt{2}}=\frac{\bold{i} + \bold{j}}{\sqrt{2}}[/tex]
Therefore, multiplying v by k(x,y) = (x+y)2/[itex]\sqrt{2}[/itex] gives the desired result.
[tex]\Rightarrow \bold{F}(x,y) = \frac{(x+y)^2}{\sqrt{2}}(\bold{i} + \bold{j})[/tex]I am just curious how other would approach this problem, or if this is the most efficient method from a mathematical standpoint.
~Casey
Homework Statement
Write the formula for a vector function in 2 dimensions whose direction makes an
angle of 45 degrees with the x-axis and whose magnitude at any point (x,y) is (x+y)2.
The Attempt at a Solution
Now just looking at it, we can say that to insure that the vector at any point (x,y) is at 45 degrees to the x-axis, we can let the funciton equal something to the effect of:
[tex]\mathbf{F}(x,y) = k(\bold{i} + \bold{j})[/tex]
Now k(x,y) is the 'scalar portion' of F and must have the effect that its product with the magnitude of the
vector portion of F, namely v=(i + j), must equal (x+y)2.
Now since the magnitude of v=(i + j) is [itex]\sqrt{2}[/itex], k(x,y) must have [itex]\sqrt{2}[/itex] in its denominator.
This will make 'unit vector' in the v direction or
[tex]\bold{u}_v = \frac{\bold{v}}{\sqrt{2}}=\frac{\bold{i} + \bold{j}}{\sqrt{2}}[/tex]
Therefore, multiplying v by k(x,y) = (x+y)2/[itex]\sqrt{2}[/itex] gives the desired result.
[tex]\Rightarrow \bold{F}(x,y) = \frac{(x+y)^2}{\sqrt{2}}(\bold{i} + \bold{j})[/tex]I am just curious how other would approach this problem, or if this is the most efficient method from a mathematical standpoint.
~Casey
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