Lorentz Force Equation: Geometric Interpretation

Somefantastik
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\textbf{L} \ = \ q\left( \textbf{v \ x \ B} \right)

L, v, and B are vectors, and B represents magnetic induction.

if \textbf{v} \ = \widehat{x},
then \textbf{v x B} = \widehat{x} \ \textbf{x B}

What is this quantity, \widehat{x} \ \textbf{x B}, geometrically?
 
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A vector orthogonal to both x and B whose magnitude corresponds to the amount of angle needed to rotate B into the x direction times the magnitude of B itself.
The magnitude is also the area of a rhombus framed by the vectors.
 
Can I find B if I know

\widehat{x} \textbf{ x B}

\widehat{y} \textbf{ x B}

\widehat{z} \textbf{ x B} ?
 

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