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Dot and Cross Product
CROSS PRODUCT OF VECTORS
Vector Product or Cross Product
If \(\overrightarrow A\) and \(\overrightarrow B\) are two vectors and the angle between them is \(\theta\) then the cross product of these two vectors is given by \( \overrightarrow A \times \overrightarrow B = \left( {AB\sin \theta } \right)\hat n \).
Where \(\hat n\) is unit vector perpendicular to the plane containing \( \overrightarrow A \,\,\& \,\overrightarrow B \).
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Dot and Cross Product
Cross product of two vectors is a vector.
* Direction is given by right hand cork screw rule or Right Hand Thumb Rule
Right handed screw rule : Imagine a right handed screw to be placed along the normal of the plane containing the two vectors. Rotate the cap of the screw from first vector to second vector through small angle between them, the direction of motion of the tip of the screw gives the direction of their vector product.
Right hand thumb rule : Imagine the normal to the plane of the two vectors to be held in the right hand with the thumb erect.
If the fingers are whirling in the direction from first vector to second vector through small angle between them, the direction of the thumb gives the direction of their vector product. -
Dot and Cross Product
Laws of Vector product:
a) Cross product does not obey commutative law.
\( \overrightarrow A \times \,\overrightarrow B \ne \,\overrightarrow B \times \overrightarrow A \) but \( \overrightarrow A \, \times \,\overrightarrow B = - \overrightarrow B \times \overrightarrow A \)
b) Cross product obeys distributive law.
\( \overrightarrow A \times \,\left( {\overrightarrow B + \overrightarrow C } \right) = \overrightarrow A \times \overrightarrow B + \overrightarrow A \times \overrightarrow C \)
c) Cross product do not obey associative law
\( \overrightarrow A \times \left( {\overrightarrow B \times \overrightarrow C } \right) \ne \left( {\overrightarrow B \, \times \overrightarrow A } \right) \times \overrightarrow C \) -
Dot and Cross Product
Properties of cross Product:
1) If \( \hat i,\hat j,\hat k \) are unit vectors then
\( \hat i \times hat i = \hat j \times \hat j = \hat k \times \hat k = \overrightarrow 0 \) , \( \hat i \times \hat j = \hat k,\hat j \times \hat k = \hat i,\hat k \times \hat i = \hat j \), \( \hat j \times \hat i = - \hat k,\hat k \times \hat j = - \hat i,\,\,\hat i \times \hat k = - \hat j \)
2) If \( \overrightarrow A = A_x \hat i + A_y \hat j + A_z \hat k \) and \( \overrightarrow B = B_x \hat i + B_y \hat j + B_z \hat k \)
then \( \overrightarrow A \times \overrightarrow B = \left| {\begin{array}{*{20}c} {\hat i} & {\hat j} & {\hat k} \\ {A_x } & {A_y } & {A_z } \\ {B_x } & {B_y } & {B_z } \\ \end{array} } \right| \) \(=(A_yB_z-A_zB_y)\hat i -(A_xB_z-A_zB_x)\hat j+(A_xB_y-A_yB_x)\hat k \)
3) Unit vector normal to both \(\overrightarrow A\) and \(\overrightarrow B\) is \( \hat n = \frac{{\overrightarrow A \times \overrightarrow B }} {{\left| {\overrightarrow A \times \overrightarrow B } \right|}} \).
4) If two vectors are parallel (\(\theta =0\)) or anti parallel (\(\theta=180^0\) ) then \( \overrightarrow A \times \overrightarrow B = 0 \)
5) If two vectors \(\overrightarrow A \) and \(\overrightarrow B\) are parallel \( \frac{{A_x }} {{B_X }} = \frac{{A_y }} {{B_y }} = \frac{{A_z }} {{B_z }} = \text{constant} \) or \( \overrightarrow A \times \overrightarrow B = 0 \)
6) If two vectors are perpendicular to each other \(\theta=90^0\) then \( \left| {\overrightarrow A \times \overrightarrow B } \right| = AB \) (maximum) -
Dot and Cross Product
Examples of cross product :
i) Angular momentum \( \mathop L\limits^ \to = \mathop r\limits^ \to \times \mathop P\limits^ \to \)
ii) Linear velocity \( \mathop V\limits^ \to = \mathop \omega \limits^ \to \times \mathop r\limits^ \to \)
iii) Torque \( \mathop \tau \limits^ \to = \mathop r\limits^ \to \times \mathop F\limits^ \to \)
iv) Torque on a magnet \( \mathop \tau \limits^ \to = \mathop M\limits^ \to \times \mathop B\limits^ \to \)
v) Force on a conductor \( \mathop F\limits^ \to = i\left( {\mathop l\limits^ \to \times \mathop B\limits^ \to } \right) \)
vi) Force on a charge \( \mathop F\limits^ \to = q\left( {\mathop v\limits^ \to \times \mathop B\limits^ \to } \right) \)
vii) Torque on a coil \( \mathop \tau \limits^ \to = i\left( {\mathop A\limits^ \to \times \mathop B\limits^ \to } \right)\)
viii) A force \(\vec F\) acts at P and \( \vec \tau \) is torque produced about Q. If position vector of P is \(\vec r_1\) and position vector of Q is \(\vec r_2\) then \( \overrightarrow \tau = \vec r\,x\,\vec F = \left( {\overrightarrow {r_2 } - \overrightarrow {r_1 } } \right)x\vec F \). -
Dot and Cross Product
Applications of cross product :
(i) The area of triangle formed by \( \overrightarrow A \& \,\overrightarrow B \) as adjacent sides is \( \frac{1} {2}\left| {\overrightarrow A \times \overrightarrow B } \right| \)
(ii) The area of parallelogram formed by \( \overrightarrow A \,\& \,\overrightarrow B \) as adjacent sides is \( \left| {\overrightarrow A \times \overrightarrow B } \right| \)
(iii) The area of parallelogram formed by \( \overrightarrow {d_1 } \& \overrightarrow {d_2 } \) as its diagonals is \( \frac{1} {2}\left| {\overrightarrow {d_1 } \times \overrightarrow {d_2 } } \right| \)