I'm reading a textbook at the moment that provides the following linear equation, α v...

minwaardekn

minwaardekn

Answered

2022-06-24

I'm reading a textbook at the moment that provides the following linear equation,
α v + v × a = b ,
and asks to solve for v . The form of v is given as
v = α 2 b α ( b × a ) + ( a b ) a α ( α 2 + | a | 2 ) .
It's easy enough to verify that this is the correct solution. However, I can't figure out how I'd solve for v if given just the original equation.
Are there any general approaches to solving this kind of equation systematically?
Edit: a , b and v are all vectors, whereas α is a scalar such that α 0.

Answer & Explanation

Colin Moran

Colin Moran

Expert

2022-06-25Added 21 answers

Taking cross product with a on both sides, we get,
α v + v × a = b α ( v × a ) + ( v × a ) × a = b × a α ( b α v ) + ( v a ) a | a | 2 v = b × a α b α 2 v + 1 α ( b a ) a | a | 2 v = b × a ( Using  α ( v a ) = b a )
Now solve for v directly.

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