# Let A = [[a, b],[c, d]] be a 2x2 matrix. Let vector v = and vector w = be the column vectors of A. Compute the cross product of vector v and w over R^3.

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Any vector `x` in `R^3` can be written as `x=x_1i+x_2j+x_3k`

where `i,j,k` are unit vectors along the three directions.

Now given matrix is `A=[[a,b],[c,d]]` . According to the problem the vector `v` is the column vector of the matrix `A` .

So we can take `v=[[a],[c]]` which can be wriiten as `v=ai+cj+0k` , a vector in `R^3.`

Now `vxxx=det[[i,j,k],[a,c,0],[x_1,x_2,x_3]]`

`=(cx_3-0x_2)i-(ax_3-0x_1)j+(ax_2-cx_1)k`

`=cx_3i-ax_3j+(ax_2-cx_1)k` . Answer.

Any vector in `R^3 ` can be written as (x,y,z).

So vector v=(a,c,0) and w=(b,d,0) are in `R^3`

`vx w =[[i,j,k],[a,c,0],[b,d,0]]`

`` `=hatk(ad-bc)`

`hati,hatj, and hatk` are mutually perpendicular directions.