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Difference between the inner product and the cross product

What is the difference between the inner product and cross product? It is difficult to explain this but I will try to do it here. The calculation of inner product is expressed as: \begin{equation} {\bf A} \cdot {\bf B} = |{\bf A}||{\bf B}| \cos \theta \end{equation} The above equation means “projecting the vector \({\bf A}\) on vector \({\bf B}\) and multiplying \(|{\bf B}|\)”. Therefore, the inner product between the vector \({\bf A}\) and vector \({\bf B}\) would be scalar.

On the other hand, the calculation of the cross product is expressed as: \begin{equation} {\bf C} = {\bf A} \times {\bf B} = | {\bf A} | | {\bf B} | \sin \theta {\bf \hat{C}} \end{equation} The vector \({\bf C}\) is perpendicular to the plane which consists of the vector \({\bf A}\) and \({\bf B}\). It is difficult to explain what the \({\bf C}\) vector is, however the cross product provides the vector. Therefore, one of the big differences between the scalar and cross products is that the solution becomes the scalar or vector.

The inner product and the cross product are very useful when calculating physical phenomena. For example, as described above, when pulling an object placed on a table, applying a force parallel to the table can move the object more efficiently when \(\cos \theta=1\). However, if you pull the object with the angle of the surface of the table, the motion efficiency decreases. The inner product is convenient to represent this sort of phenomena.

The best example of the use of the outer product is for the Lorenz’s force. The Lorentz’s force on the scalar is expressed as: \begin{eqnarray} |{\bf F}| = q |{\bf v}| |{\bf B}| \sin \theta \end{eqnarray} where \( {\bf F} \) is the force, \(q\) is the electric charge, \({\bf v}\) is the velocity of the charge, and \({\bf B}\) is the magnetic field. \(\theta \) is the angle between \({\bf v}\) and \({\bf B}\). In this case, the direction \({\bf F}\) is perpendicular to both \({\bf v}\) and \({\bf B}\). The Lorenz’s force can be expressed using the vectors \({\bf v}\) and \({\bf B}\). \begin{eqnarray} {\bf F} = q {\bf v} \times {\bf B} \end{eqnarray} Thus, an equation such as the Lorenz’s force can be simplified by the cross product.

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