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Considering that I have a uniform magnetic field given by $\vec B = B_0 \hat k$ and $\vec v = (a\hat i,b\hat j,c\hat k )$.

Taking into account that the radius of the spiral path can be acquired by $R = \frac{mv}{qB}$, what are the components of the velocity vector that i should consider for the calculation? I believe that I should only consider the components in x and y, since the field points in the z direction. However, I saw some resolutions for these types of problems that consider all the velocity components in the calculation. Thanks in advance.

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You are right that z component of velocity remains unchanged. But this is taken care of automatically when you calculate the Lorentz force using vector notation: $$\vec F= Q\vec v \times \vec B = Q(a \hat i+b\hat j +c\hat k)\times B_0 \hat k=QB_0(-a \hat j+b\hat i)$$ We have used the fact that $\hat k \times \hat k =0$. So there is no force component on Q in the z direction, as you predicted. That means that the circular motion takes place in a plane at right angles to the field, so only the velocity components in that plane are relevant. Note that your $a$ and $b$ are, in fact, variables.

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  • $\begingroup$ So can I use it $v = \sqrt{(a \hat i)^2 + (b \hat j)^2 + (c \hat k)^2}$ for the radius formula? Like $R = \frac{mv^2}{F}$. $\endgroup$
    – Jacobi
    Commented Oct 18, 2020 at 22:46
  • $\begingroup$ No. I'm sorry that I didn't understand originally what you were asking. I've added a bit to my answer, and might edit it again the next day or two. $\endgroup$
    – Philip Wood
    Commented Oct 18, 2020 at 23:09

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