Solving For Two Vectors With a Given Dot and Cross Product

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Given vectorsandwith

we can solve these equations by first taking the dot product of (1) wuthto give

The left hand side of this equation equals zero sinceis perpendicular toso

Now use the identity

Substituteandto give

Henceandare multiples of each other or

Ifandare multiples of each other, from (1)andare pendicular so both are 0 then from (2),is arbitrary.

Ifthen from (1),andare parallel so