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Define Unit Vectors In Planer Motion In Terms Of Their Cartesian Counter-Parts.

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In a two-dimensional plane polar co-ordinate system, there are two unit vectors \cdots

  1. The radial unit vector   \hat{r} . The direction of this unit vector is along the radius vector   \vec{r}
  2. The unit vector   \hat{\theta} along the direction of increasing \theta .

Similarly, in case of planer motion in a two-dimensional Cartesian co-ordinate system, there are also two unit vectors,

  1. The unit vector \hat{i} along the direction of the X-axis.
  2. The unit vector \hat{j} along the direction of the Y-axis

Relations:

  • The unit vector \hat{r} in term of the unit vectors \hat{i} and \hat{j} :
    Let, In two dimensional cartesian co-ordinate system, (x,y) be the position vector of the point A
    And, in plane polar co-ordinate system, ( \hat{r} , \theta ) be the position vector of the same point A , as shown in the figure 1.1.
    Here, \vec{OA}=\vec{r} .
    So the unit vector \hat{r} is along the direction of the vector \vec{r} , i.e., \vec{OA} . The unit vector \hat{\theta} is perpendicular to the vec{OA} .
    The unit vector \displaystyle{\hat{r}=\frac{\vec{r}}{|\vec{r}|}} , where, |\vec{r}| is the modulus of the vector \vec{r} .
    Now, \vec{r}=x\hat{i}+y\hat{j}
    Again, x=r\cos\theta & y=r\sin\theta
    therefore, \vec{r}=r\cos\theta\hat{i}+r\sin\theta\hat{j}
    so the unit vector, \displaystyle{\hat{r}=\frac{\vec{r}}{|\vec{r}|}=\frac{ r\cos\theta\hat{i}+r\sin\theta\hat{j}}{r}}
    or, \hat{r}=\cos\theta\hat{i}+\sin\theta\hat{j} .
Fig. 1.1
  • The unit vector \hat{\theta} in term of the unit vectors \hat{i} and \hat{j} :
    Let us consider a point A having polar co-ordinate (r,\theta) & another point B having polar co-ordinate (r,\theta+d\theta) , as shown in the adjoining figure 1.2.
    \vec{OA}=r\cos\theta\hat{i}+r\sin\theta\hat{j}
    and, \vec{OB}=r\cos(\theta+d\theta)\hat{}+r\sin(\theta+d\theta)\hat{j}
    Again, \vec{AB}=\vec{OA}-\vec{OB} .
    or, \vec{AB}= r\cos\theta\hat{i}+r\sin\theta\hat{j}- r\cos(\theta+d\theta)\hat{}+r\sin(\theta+d\theta)\hat{j}
    or, \vec{AB}=r[\cos(\theta+d\theta)-\cos\theta]\hat{i}+r[\sin(\theta+d\theta)-\sin\theta]\hat{j}
    or \vec{AB}=-r\ \sin\theta\ d\theta\hat{i}+r\ \cos\theta\ d\theta\hat{j}

    Where, \cos(\theta+d\theta)-\cos\theta=\cos\theta\ \cos{d\theta}-\sin\theta\ \sin{d\theta}-\cos\theta
    or, \cos(\theta+d\theta)-\cos\theta=-\sin\theta\ d\theta
    [ Since, d\theta is very small. So \cos{d\theta}=1 and \sin{d\theta}=d\theta ]
    Similarly, \sin(\theta+d\theta)-\sin\theta=\sin\theta\ \cos{d\theta}-\cos\theta\ \sin{d\theta}-\sin\theta
    or, \sin(\theta+d\theta)-\sin\theta =\cos\theta\ d\theta

    The direction of the vector \vec{AB} is along the direction of the increasing \theta .
    The magnitude of the vector \vec{AB} is,
    |\vec{AB}|=r\ d\theta\sqrt{({\sin}^2\theta+{\cos}^2\theta)}=r\ d\theta
    The unit vector
    \displaystyle{\hat{\theta}=\frac{\vec{AB}}{|\vec{AB}|}}
    or, \displaystyle{\hat{\theta}=\frac{-r\ \sin\theta\ d\theta\hat{i}+r\ \cos\theta\ d\theta\hat{j}}{r\ d\theta}}
    or, \hat{\theta}=-\sin\theta\hat{i}+\cos\theta\hat{j}
Fig. 1.2

The polar unit vectors vary from point to point, they are not fixed like \hat{i} and \hat{j} . When a point moves, the polar unit vectors become a function of time.

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