Vector products

$a*b={\displaystyle \underset{i=1}{\overset{n}{}}}(a_i*b_i)$

Here $a_i$ and $b_i$ are the ${?}^{th}$ element of the vector and $n$ being the dimension of the vector.

Use the function to verify that $[2,6,2]*[2,1,-1]$ returns $8.$

$($b$)$ Now, create a function that takes two vectors of dimension three in list or tuple form and returns the cross product. The cross product for two three length vectors, $a=a_{x}i+a_{y}j+a_{z}k,$ and $b=b_{x}i+b_{y}j+b_{z}k$ is given by:

$ab=(a_y{b}_z-a_z{b}_y)i+(a_z{b}_x-a_x{b}_z)j+(a_x{b}_y-a_y{b}_x)k$

Verify the two lists from above return the list $-8,6,-10$ and show the formatted string $-8i+6j-10k$ on the screen.

$A_{ij}=a_i{b}_j$

Here $a_i$ and $b_j$ are the $i^{th}$ and $j^{th}$ elements of the vectors vec$(a)$ and vec$(b),$ while $A_{ij}$ is the element of the resulting matrix in row i and column $).$

For the vectors in the previous two examples, the outer product in a nested list form will be:

$[[4,2,-2]$

$[12,6,-6]$

$[4,2,-2]]$