Express VectorU in Terms of VectorA, VectorB, ScalarF

[starbaj12]

vector U is an unkown vector; known vectors are vectorA and vectorB and scalarF. vectorA (cross) vectorU = vectorB and vectorA (dot) vectorU = F. Express vectorU in terms of vectorA, vectorB, scalarF and the magnitude of vectorA.

Thank you

 

[mathwonk]

given A,B,f, such that AxU = B, and A.U = f, find U in terms of A,B,f.

well let's see what we have. AxU = B means that B is perpendicular to the plane spanned by A and U. A.U = f, means that |A||U|cos(t) = f where t is the angle between A and U. Then B/|B| x A/|A| = C is a unit vector perpendicular to the plane spanned by A and B, hence lies in the plane spanned by A and U. So U is in the plane spanned by A and C, which are perpendicular to each other, but the angle t between A and U has cosine equal to f/(|A||U|. So C and A/|A| =V are orthonormal vectors and U is a linear combination of them. Indeed U/|U| = cos(t)A + sin(t)C.

now all we need is the length of U. But |B| = |A||U| sin(t), so |U| = |B|/|A|sin(t).

that is pretty close if you note that sin(t) is positive, hence determined by cos(t).

does that help?

 

[starbaj12]

Thank you for your reply,

But could you elaborate on, "that is pretty close if you note that sin(t) is positive, hence determined by cos(t)"

Thank you for your help

 

[mathwonk]

i mean sin(t) = +sqrt(1-cos^2(t)), so everything is known.