Why does $\vec v \times \vec B$ change to $\vec a \cdot \vec B$? [closed]

Question

I was doing a question where the solution said this:

$$\vec F = q (\vec v \times \vec B)$$

$$m \vec a= q (\vec v \times \vec B)$$

$$ \therefore \vec a \cdot \vec B = 0 $$

$$(x\hat i + 7 \hat j) \cdot (7 \hat i - 3 \hat j)=0 $$

$$7x-21=0$$

$$x=3$$

Please explain why does the $\vec v \times \vec B$ turn into $\vec a \cdot \vec B$, should it be $\vec a \times \vec B$ cause like $\vec v$ & $\vec a$ are also perpendicular to $\vec B$

Answer 1

It's a property of vectors.

The cross product of two vectors $\vec v \times \vec B$ will always be perpendicular to both of the original vectors $\vec v$ and $\vec B$.

Because of this $\vec a$ is perpendicular to $\vec B$. This in turn implies that the dot product $\vec a \cdot \vec B$ is zero.