How to solve the equation and directly determine the value of vec(v) in the equation? vec(v)-a[vec(v) xx vec(y)]=b vec(E) where, vec(v) and vec(E) are in the hat(x) direction, and a and b are scalars.

mydaruma25 2022-09-23 Answered
How to solve the equation and directly determine the value of v in the equation?
v a [ v × y ^ ] = b E
where, v and E are in the x ^ direction, and a and b are scalars.
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Answers (2)

Miguel Shah
Answered 2022-09-24 Author has 8 answers
You can convert c = a × b into a matrix-vector product with the following trick (it is called the cross product operator matrix).
c = [ a × ] b [ a × ] = ( 0 a z a y a z 0 a x a y a x 0 )
So the LHS of the equation above is
v a ( v × y ^ ) = v + a ( y ^ × v ) = ( 1 + a [ y ^ × ] ) v
where 1 is 3×3 the identity matrix, and [ y ^ × ] the 3×3 cross product operator.
This makes an equation like v⃗ −a(v⃗ ×y^)=bE⃗ v a ( v × y ^ ) = b E solvable
v = ( 1 + a [ y ^ × ] ) 1 b E
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Hana Buck
Answered 2022-09-25 Author has 2 answers
Given that v is in the x-direction, ( v x 0 0 ) × ( 0 1 0 ) = ( 0 0 v x ) = c . To find v , solve for the components algebraically and separately. In this case, that’s just v x ; if v is in the x-direction, then v y = v z = 0
v x a c x = b E x v x = b E x + a c x = b E x + 0
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