Main Concept

The cross product of $\stackrel{\mathbf{\rightharpoonup }}{\mathit{A}}$ and $\stackrel{\mathbf{\rightharpoonup }}{\mathit{B}}$  is a vector denoted $\stackrel{\mathbf{\rightharpoonup }}{\mathit{A}}$ × $\stackrel{\mathbf{\rightharpoonup }}{\mathit{B}}$. The magnitude of $\stackrel{\rightharpoonup }{A} \times \stackrel{\rightharpoonup }{B }$ is given by

The direction of $\stackrel{\rightharpoonup }{A} \times \stackrel{\rightharpoonup }{B }$ is perpendicular to the plane formed by $\stackrel{\rightharpoonup }{A}$ and $\stackrel{\rightharpoonup }{B}$, and obeys the right hand rule:

Position the middle and index fingers and thumb of your right hand at right angles with the index finger pointing straight. If the middle and index fingers approximate the directions of vectors $\stackrel{\mathit{\rightharpoonup }}{A}$ and $\mathit{ }\stackrel{\mathit{\rightharpoonup }}{B\mathit{ }}$ respectively, then the thumb will be in the direction of $\stackrel{\mathit{\rightharpoonup }}{A}\mathit{ }\mathit{\times }\mathit{ }\stackrel{\mathit{\rightharpoonup }}{B\mathit{ }}$.

The components of cross product $\stackrel{\rightharpoonup }{C}$ are expressed in terms of those for $\stackrel{\rightharpoonup }{A}$ and $\stackrel{\rightharpoonup }{B}$ as follows:

$C_x \= A_y\cdot B_z- A_z\cdot B_y$

$C_y\=A_z \cdot B_x - A_x\cdot B_z$

$C_z \= A_x\cdot B_y- A_y\cdot B_x$

More compactly, the cross product can be written using a determinant:

$\stackrel{\rightharpoonup }{A} \times \stackrel{\rightharpoonup }{B } \= \left[\begin{array}{ccc}\overbrace{\mathbf{i}}& \overbrace{\mathit{j}}& \overbrace{\mathit{k}}\\ A_x& A_y& A_z\\ B_x& B_y& B_z\end{array}\right]$

where, $\overbrace{\mathbf{i}}$, $\overbrace{\mathit{j}}$, and $\overbrace{\mathit{k}}$ are unit vectors in the $\mathit{x}$, $\mathit{y}$, and $\mathit{z}$ directions respectively.

Note:       $\stackrel{\rightharpoonup }{A} \times \stackrel{\rightharpoonup }{B } \= -\stackrel{\rightharpoonup }{B} \times \stackrel{\rightharpoonup }{A}$ (that is, the vector cross products are anticommutative).

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