$\hat{\imath}$ component of force exerted on an electron by a magnetic field?

The magnetic field over a certain range is given by $\vec{B} = B_x\hat{\imath} + B_y\hat{\jmath}$, where $B_x= 4\: \mathrm{T}$ and $B_y= 2\: \mathrm{T}$. An electron moves into the field with a velocity $\vec{v} = v_x\hat{\imath}+v_y\hat{\jmath}+v_z\hat{k}$, where $v_x= 5\: \mathrm{m/s}$, $v_y= 8\: \mathrm{m/s}$ and $v_z= 9\: \mathrm{m/s}$. The charge on the electron is $-1.602 \times 10^{-19}\: \mathrm{C}$. What is the $\hat{\imath}$ component of the force exerted on the electron by the magnetic field? Answer in units of $\mathrm{N}$.

I know that $\vec{F}=q\vec{v} \times \vec{B}$, so plugging in I have:

$$\vec{F}=(-1.602 \times 10^{-19})<5,8,9> \times <4,2,0>$$

My confusion is to whether or not multiply my velocity vector components by charge (the scalar) or if I take the cross product between $\vec{B}$ and $\vec{v}$ first? I'd also like to know why whichever operation comes first does in fact come first.

• Your question is not actually dependent on the homework problem and is more general than that. You will be more likely to get a good response - and also avoid your question being closed as homework - if you remove the homework assignment and concentrate on your conceptual question. – Emilio Pisanty Dec 11 '13 at 14:39
• I didn't originally have the homework tag, just the e/m tag. Someone edited it I guess, but i'll go ahead and take it out, thank you! – Lame-Ov2.0 Dec 11 '13 at 18:33
• To be clear, I do not mean the homework tag, which is indeed appropriate at the current state of the question. I mean you should cut out all reference to the fact that this is grounded on a homework question. That makes the core concept ("do I multiply by a scalar before or after the vector product?") a lot clearer. – Emilio Pisanty Dec 11 '13 at 19:04

1 Answer

It doesn't matter. If you have a scalar $\alpha$ then

$$\alpha (\vec{B} \times \vec{C}) = (\alpha \vec{B}) \times \vec{C} = \vec{B} \times (\alpha \vec{C}).$$

You can prove this simply by writing out the components of each of the three expressions and showing that no matter which order you do it in you will get the same results.