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Math Help - Vector calculus

  1. #1
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    Vector calculus

    If T=ai+bj

    since N is orthogonal to T and both has length 1,

    so I set

    N=bj-ai or N= -bi+aj (which is correct)

    How can I make sure that N(s) is in the direction as dT/ds

    (dT/ds points in the direction in which T turns as the curve bends)
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  2. #2
    Moo
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    Hello,

    The derivative of the unit vector \vec{i} is related to \vec{j} and vice-versa :

    \frac{d\vec{i}}{dt}=\vec{j}

    \frac{d \vec{j}}{dt}=-\vec{i}

    This may help you
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    Quote Originally Posted by Moo View Post
    Hello,

    The derivative of the unit vector \vec{i} is related to \vec{j} and vice-versa :

    \frac{d\vec{i}}{dt}=\vec{j}

    \frac{d \vec{j}}{dt}=-\vec{i}

    This may help you
    I still don't understand.



    And which N is correct.
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  4. #4
    Moo
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    Oh i thought you were saying that the two were correct.

    Well, for me it's N= -bi+aj which is correct.

    How can I make sure that N(s) is in the direction as dT/ds
    Derivate T.

    \vec{T}=a \vec{i}+b \vec{j}

    As a and b are constant, we have :

    \frac{d \vec{T}}{ds}=a \frac{d \vec{i}}{ds} + b \frac{d \vec{j}}{ds}

    According to what i wrote above :

    \frac{d \vec{T}}{ds}=a \vec{j} - b \vec{i}, which is the same direction as N.
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    Quote Originally Posted by Moo View Post
    Oh i thought you were saying that the two were correct.

    Well, for me it's N= -bi+aj which is correct.



    Derivate T.

    \vec{T}=a \vec{i}+b \vec{j}

    As a and b are constant, we have :

    \frac{d \vec{T}}{ds}=a \frac{d \vec{i}}{ds} + b \frac{d \vec{j}}{ds}

    According to what i wrote above :

    \frac{d \vec{T}}{ds}=a \vec{j} - b \vec{i}, which is the same direction as N.

    a and b are not constants
    a and b are functions of variable t
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  6. #6
    Moo
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    Not of s though... So can it be considered as constant if the variable is s ?

    What's the exact text ?
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  7. #7
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    Quote Originally Posted by Moo View Post
    Not of s though... So can it be considered as constant if the variable is s ?

    What's the exact text ?
    consider this question

    r(t)=3tcos(t)j-3tsin(t)k

    determine N,T,B

    you can get the answer from this website

    default

    Section 12-2
    Problem 10

    and you'll see that if
    T=aj+bk then N=bj-ak
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  8. #8
    Moo
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    Outch, my eyes >.<

    Well, if i got the pitch, T is the unit vector of V, which you can get by dividing V by its norm.
    N is the normal vector of T, that is to say the derivative of T in t.

    But you can also remember that if you consider a vector of (a,b) coordinates, one of its normal vector will have (b,-a) coordinates (scalar product). Another of its normal vectors will have (-a,b) coordinates. Never mind, the direction remains the same, it's just the sens that will change.

    This explains the formula N=bj-ai or N=-bj+ai

    Then for this :

    How can I make sure that N(s) is in the direction as dT/ds
    I need more time... but it's not the thing i prefer to do :s
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  9. #9
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    Quote Originally Posted by Moo View Post
    Outch, my eyes >.<

    Well, if i got the pitch, T is the unit vector of V, which you can get by dividing V by its norm.
    N is the normal vector of T, that is to say the derivative of T in t.

    But you can also remember that if you consider a vector of (a,b) coordinates, one of its normal vector will have (b,-a) coordinates (scalar product). Another of its normal vectors will have (-a,b) coordinates. Never mind, the direction remains the same, it's just the sens that will change.

    This explains the formula N=bj-ai or N=-bj+ai

    Then for this :



    I need more time... but it's not the thing i prefer to do :s
    The question you do not prefer is exactly the question I asked at the first time.

    There can be only one N ,which is in the same direction as T'(s).

    and I wonder which N is correct
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  10. #10
    Moo
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    Well, at the beginning, it was not clear about what was a, what was b. I don't understand either the variable is s or t...

    I find strange things for T'(t)...
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  11. #11
    Moo
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    Ok, i was not looking at the right function... I took V on the website you gave.

    r(t)=3tcos(t)j-3tsin(t)k
    What do you name N, T and B ??? (just to be sure...)
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