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Show That The Spherical Polar Unit Vectors Are Mutually Perpendicular To Each Other.

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Unit vectors in spherical polar co-ordinate system are mutually perpendicular to each other:

Two unit vectors are said to be orthogonal to each other when the angles between these two unit vectors are 90^{\circ} , i.e., they are perpendicular to each other.

In order to prove the unit vectors ( \hat{r} , \hat{\theta} , \hat{\phi} ) in spherical polar co-ordinate system are mutually perpendicular to each other, we have to prove that the angles between \hat{r} & \hat{\theta} , \hat{\theta} & \hat{\phi} , \hat{r} & \hat{\phi} are 90^{\circ} each.

Let us consider a point A having spherical polar co-ordinate ( r,\theta,\phi ) as shown in figure 1.

Fig.1

The unit vectors in three-dimensional polar co-ordinate system are \hat{r} , \hat{\theta} , \hat{\phi} .

So we can write,

\hat{r}=\sin\theta\ \cos\phi\ \hat{i}+\sin\theta\ \sin\phi\ \hat{j}+\cos\theta\ \hat{k}

\hat{\theta}=\cos\theta\ \cos\phi\ \hat{i}+\cos\theta\ \sin\phi\ \hat{j}-\sin\theta\ \hat{k}

\hat{\phi}=-\sin\phi\ \hat{i}+\cos\phi\ \hat{j}

where, \hat{i}.\hat{j},\hat{k} are the unit vectros in the three-dimensional cartesian co-ordinate system along X-axis, Y-axis and Z-axis respectively.

[ To know the derivation of the above unit vectors, CLICK HERE ]

Angle between \hat{r} & \hat{\theta} :

The scaler product of \hat{r} & \hat{\theta} is given by,

\hat{r}\cdot\hat{\theta}=(\sin\theta\ \cos\phi\ \hat{i}+\sin\theta\ \sin\phi\ \hat{j}+\cos\theta\ \hat{k})\cdot(\cos\theta\ \cos\phi\ \hat{i}+\cos\theta\ \sin\phi\ \hat{j}-\sin\theta\ \hat{k})

or, \hat{r}\cdot\hat{\theta}=\sin\theta\ \cos\theta\ {\cos}^2\phi+\sin\theta\ \cos\theta\ {\sin}^2\phi-\cos\theta\ \sin\theta

or, \hat{r}\cdot\hat{\theta}=\sin\theta\ \cos\theta({\cos}^2\phi+{\sin}^2\phi)-\sin\theta\ \cos\theta

or, \hat{r}\cdot\hat{\theta}=\sin\theta\ \cos\theta-\sin\theta\ \cos\theta

or, \hat{r}\cdot\hat{\theta}=0

[where, \hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{i}\cdot\hat{k}=0 and \hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1 ]

So the angle between \hat{r} and \hat{\theta} is 90^{\circ} , i.e., the unit vectors \hat{r} and \hat{\theta} are perpendicular to each other.

Angle between \hat{\theta} & \hat{\phi} :

The scalar product of the unit vectors \hat{\theta} and \hat{\phi} is given bellow,

\hat{\theta}\cdot\hat{\phi}=(\cos\theta\ \cos\phi\ \hat{i}+\cos\theta\ \sin\phi\ \hat{j}-\sin\theta\ \hat{k})\cdot(-\sin\phi\ \hat{i}+\cos\phi\ \hat{j})

or, \hat{\theta}\cdot\hat{\phi}=-\cos\theta\ \sin\phi\ \cos\phi+\cos\theta\ \sin\phi\ \cos\phi

or, \hat{\theta}\cdot\hat{\phi}=0

So the angle between the unit vectors \hat{\theta} & \hat{\phi} is   90^{\circ} , i.e., the unit vectors \hat{\theta} & \hat{\phi} are perpendicular to each other.

Angle between \hat{r} & \hat{\phi} :

The scalar product of the unit vectors \hat{r} and \hat{\phi} is given bellow,

\hat{r}\cdot\hat{\phi}=(\sin\theta\ \cos\phi\ \hat{i}+\sin\theta\ \sin\phi\ \hat{j}+\cos\theta\ \hat{k})\cdot(-\sin\phi\ \hat{i}+\cos\phi\ \hat{j})

or, \hat{r}\cdot\hat{\phi}=-\sin\theta\ \sin\phi\ \cos\phi+\sin\theta\ \sin\phi\ \cos\phi

or, \hat{r}\cdot\hat{\phi}=0

So the angle between the unit vectors \hat{r} & \hat{\phi} is 90^{\circ} , i.e., the unit vectors \hat{r} & \hat{\phi} are perpendicular to each other.

Therefore the unit vectors \hat{r} , \hat{\theta} and \hat{\phi} in the spherical polar co-ordinate system are mutually perpendicular to each other.

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