Productos vectoriales

[natanael]

$$\dst \hat{\imath} \otimes \hat{\imath}=\hat{\jmath} \otimes \hat{\jmath} =\hat{k} \otimes \hat{k} =0$$

$$\dst \hat{\imath} \otimes \hat{\jmath} =\hat{k}$$

$$\dst \hat{\imath} \otimes \hat{k} =-\hat{\jmath}$$

$$\dst \hat{\jmath} \otimes \hat{\jmath} =-\hat{k}$$

$$\dst \hat{\jmath} \otimes \hat{k} =\hat{\imath}$$

$$\dst \hat{k} \otimes \hat{\imath} =\hat{\jmath}$$

$$\dst \hat{k} \otimes \hat{\jmath} =-\hat{\imath}$$

$\dst (a_x\hat{\imath} +a_y\hat{\jmath} +a_z\hat{k} )(b_x\hat{\imath} +b_y\hat{\jmath} +b_z\hat{k} )=a_xb_x(\hat{\imath} \otimes \hat{\imath} )+a_xb_y(\hat{\imath} \otimes \hat{\jmath} )+a_xb_z(\hat{\imath} \otimes \hat{k} )+$

$\dst a_yb_x(\hat{\jmath} \otimes \hat{\imath} )+a_yb_y(\hat{\jmath} \otimes \hat{\jmath} )+a_yb_z(\hat{\jmath} \otimes \hat{k} )+a_zb_x(\hat{k} \otimes \hat{\imath} )+a_zb_y(\hat{k} \otimes \hat{\jmath} )+a_zb_z(\hat{k} \otimes \hat{k} )$


$\dst (a_x\hat{\imath} +a_y\hat{\jmath} +a_z\hat{k} )(b_x\hat{\imath} +b_y\hat{\jmath} +b_z\hat{k} )=a_xb_y\hat{k} -a_xb_z\hat{\jmath}-a_yb_x\hat{k}+a_yb_z\hat{\imath} +a_zb_x\hat{\jmath} -a_zb_y\hat{\imath}$


$\dst (a_x\hat{\imath} +a_y\hat{\jmath} +a_z\hat{k} )(b_x\hat{\imath} +b_y\hat{\jmath} +b_z\hat{k} )=(a_yb_z-a_zb_y)\hat{\imath}+(a_zb_x - a_xb_z)\hat{\jmath} +(a_xb_y-a_yb_x)\hat{k}$




$\dst (a_x\hat{\imath} +a_y\hat{\jmath} +a_z\hat{k} )(b_x\hat{\imath} +b_y\hat{\jmath} +b_z\hat{k} )=\begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$

$\dst (a_x\hat{\imath} +a_y\hat{\jmath} +a_z\hat{k} )(b_x\hat{\imath} +b_y\hat{\jmath} +b_z\hat{k} )=\begin{vmatrix}a_y & a_z \\ b_y & b_z \end{vmatrix}\hat{\imath} -\begin{vmatrix}a_x & a_z \\ b_x & b_z \end{vmatrix}\hat{\jmath} +\begin{vmatrix}a_x & a_y \\ b_x & b_y \end{vmatrix}\hat{k}$

$\dst (a_x\hat{\imath} +a_y\hat{\jmath} +a_z\hat{k} )(b_x\hat{\imath} +b_y\hat{\jmath} +b_z\hat{k} )=(a_yb_z-a_zb_y)\hat{\imath}-(a_xb_z-a_zb_x)\hat{\jmath} +(a_xb_y-a_yb_x)\hat{k}$

$\dst (a_x\hat{\imath} +a_y\hat{\jmath} +a_z\hat{k} )(b_x\hat{\imath} +b_y\hat{\jmath} +b_z\hat{k} )=(a_yb_z-a_zb_y)\hat{\imath}+(a_zb_x - a_xb_z)\hat{\jmath} +(a_xb_y-a_yb_x)\hat{k}$