# How to rewrite a vector cross product? [closed]

Can this vector cross product $\vec F = q \vec v \times \vec B$ be rewritten to $\vec B = \vec F \times q \vec v$ ?

Edit: Take the special case, that all three vectors are orthogonal to each other.

## closed as off-topic by Adam Hughes, user99914, Chappers, Daniel W. Farlow, NewbApr 4 '15 at 4:19

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In fact the vector triple product identity says

$$(\vec {qv} \times \vec{B}) \times \vec{qv} = \|\vec{qv}\|^2 \vec B - (\vec{qv} \cdot \vec B) \vec{qv}$$

Assuming $\vec{qv}$ and $\vec{B}$ are linearly independent, this is $\vec B$ if and only if $\vec{qv}$ is a unit vector orthogonal to $\vec B$.

EDIT: In particular, if $\vec{qv}$ is orthogonal to $\vec B$, and is nonzero (but not necessarily a unit vector), you can write

$$\vec B = \dfrac{\vec F \times \vec{qv}}{\|\vec{qv}\|^2}$$

$$F = qvB,\ B = \frac{F}{qv}$$
However, you are taking the cross products of vectors, which isn't a scalar quantity, it is a vector quantity. The direction of $F$ is determined using the right-hand rule. You cannot determine the direction of $B$ given the vectors $F$ and $v$. The only thing you can do is find the scalar value of $B$ by the formula above.