Let’s beginning by discovering the cross item $latex vec {} times vec {B} $:
$ latex vec {} times vec {B} =(( − 3)( − 3) − (2 )( 1 )) hat {i} + (( 1 )( 1) − (− 3)( 2 )) hat {j} ++(( 2 )( 2) − (1 )( − 3)) hat {k} $
$ latex =7hat {i} +7 hat {j} + 7hat {k} $
To show that $latex vec {} times vec {B} $ is vertical to $latex vec {} $, we should have $latex (vec {} times vec {B} )cdot vec {} =0$:
$$( vec {} times vec {B} )cdot vec {} =start {pmatrix} 7 \ 7 \ 7 end {pmatrix} cdot start {pmatrix} 2 \ -3 \ 1 end {pmatrix} $$
$ latex = 14 − 21 + 7 = 0$
To program that $latex vec {} times vec {B} $ is vertical to $latex vec {B} $, we should have $latex (vec {} times vec {B} )cdot vec {B} =0$:
$$( vec {} times vec {B} )cdot vec {B} =start {pmatrix} 7 \ 7 \ 7 end {pmatrix} cdot start {pmatrix} 1 \ 2 \ -3 end {pmatrix} $$
$ latex = 7 +14 -21 = 0$