Arturo O. answered • 04/02/18
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I think your confusion is regarding the magnitude of the polar unit vector R. It is exactly 1 regardless of θ, as shown below.
|R| = |cosθ i + sinθ j| = √(cos2θ + sin2θ) = √1 = 1
Arturo O.
I just showed you in my answer that |R| = 1 for all θ. Do you know how to find the magnitude of a vector? Are you familiar with the trigonometric identity
cos2θ + sin2θ = 1
for all θ? Apply these two, and you will see that |R| = 1 for all θ. The equation in your Comment is not the correct equation for the magnitude of the vector R. I gave you the correct equation in my Answer.
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04/02/18
Arturo O.
Your equation for the magnitude of R is not correct. I gave you the correct equation in my Answer. Evaluate the magnitude of R using the correct equation and you will see that it gives 1 for all θ.
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04/02/18
Callista H.
I think I understand now. The source of my confusion was that I was not adding them like vectors should be added. According to the equation provided by my MIT OpenCoursework textbook:
"The unit vectors (r,θ) at the point P also are related to the Cartesian unit vectors (i, j)
by the transformations
r = cosθi + sinθj , (3.2.6)
θ = −sinθi + cosθj . (3.2.7) "
Which is true. My problem is that I was using it as though you could add vectors like any other number. (see above 1=cos(45)*1+sin(45)*1 ). Using that equation correctly, I can derive the equation you provided when I add those vectors via tail-to-tip method.
/|
r/ |sinθj=sinθ*1=sinθ **r2=cos2θ+sin2θ
/_| **r=√(cos2θ+sin2θ)
cosθi=cosθ*1=cosθ
"The unit vectors (r,θ) at the point P also are related to the Cartesian unit vectors (i, j)
by the transformations
r = cosθi + sinθj , (3.2.6)
θ = −sinθi + cosθj . (3.2.7) "
Which is true. My problem is that I was using it as though you could add vectors like any other number. (see above 1=cos(45)*1+sin(45)*1 ). Using that equation correctly, I can derive the equation you provided when I add those vectors via tail-to-tip method.
/|
r/ |sinθj=sinθ*1=sinθ **r2=cos2θ+sin2θ
/_| **r=√(cos2θ+sin2θ)
cosθi=cosθ*1=cosθ
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04/03/18
Arturo O.
Very good, Callista
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04/03/18
Callista H.
1=cos(45)*1+sin(45)*1
04/02/18