Dot and Cross Product

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Dot Product

$C = \overset{\rightharpoonup}{A}\cdot \overset{\rightharpoonup}{B} =\:AB\cos\phi$ 

With $\phi$ being the angle in between $\overset{\rightharpoonup}A$ and $\overset{\rightharpoonup}B$.

If

$\overset{\rightharpoonup}{A}=A_x\hat{i} +A_y\hat{j} +A_z\hat{k}$

$\overset{\rightharpoonup}{B}=B_x\hat{i} +B_y\hat{j} +B_z\hat{k}$

$C=\overset\rightharpoonup{A}\cdot \overset\rightharpoonup{B}= (A_x\hat{i} +A_y\hat{j} +A_z\hat{k}) \cdot (B_x\hat{i} +B_y\hat{j} +B_z\hat{k})$

$C= A_xB_x +A_y B_y +A_zB_z$

The Dot Product results in a scalar, thus the alternative name, Scalar Product.

The scalar product obeys the distributive law of multiplication therefore,

$\overset{\rightharpoonup}{A}\cdot (\overset{\rightharpoonup}{B} + \overset{\rightharpoonup}{C})=\overset{\rightharpoonup}{A}\cdot \overset{\rightharpoonup}{B} + \overset{\rightharpoonup}{A}\cdot \overset{\rightharpoonup}{C}$

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Cross Product

$\overset{\rightharpoonup}{C} = \overset{\rightharpoonup}{A}\times \overset{\rightharpoonup}{B}$

$C=\:AB\sin\phi$

With $\phi$ being the angle in between $\overset{\rightharpoonup}A$ and $\overset{\rightharpoonup}B$, the equation above only results in the magnitude of the cross product.

If,

$\overset{\rightharpoonup}{A}=A_x\hat{i} +A_y\hat{j} +A_z\hat{k}$

$\overset{\rightharpoonup}{B}=B_x\hat{i} +B_y\hat{j} +B_z\hat{k}$

$\overset{\rightharpoonup}{A}\times\overset{\rightharpoonup}{B} = \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y &A_z \\ B_x & B_y & B_z \end{matrix}=\begin{matrix} A_y & A_z \\ B_y & B_z \\ \end{matrix} \hat{i} - \begin{matrix} A_x & A_z \\ B_x & B_z \\ \end{matrix} \hat{j} +\begin{matrix} A_x & A_y \\ B_x & B_y \\ \end{matrix} \hat{k}$

Expanded form of the determinant below,

$\overset{\rightharpoonup}{A}\times\overset{\rightharpoonup}{B} =(A_yB_z-A_zB_y)\hat{i}-(A_xB_z-A_zB_x)\hat{j}+(A_xB_y-A_yB_x)\hat{k}$

The Cross Product results in a vector, thus the alternative name, Vector Product.

The vector product also obeys the distributive law.

$\overset{\rightharpoonup}{A}\times (\overset{\rightharpoonup}{B} + \overset{\rightharpoonup}{C})=\overset{\rightharpoonup}{A}\times \overset{\rightharpoonup}{B} + \overset{\rightharpoonup}{A}\times \overset{\rightharpoonup}{C}$