Showing particle travels at constant speed (geometry)

[jack1990]

Homework Statement

the trajectory γ: ℝ→ℝ³ of a charged particle moving in a uniform magnetic field satisfies the differential equation γ^''= B x γ^'(t) . where B = (B1, B2, B3) is a constant 3-vector describing the magnetic field, and × denotes the vector product.

(a) Show that the particle travels at constant speed.
(b) Show that the component of the particle’s velocity in the direction of the magnetic field B is also constant.

Homework Equations

The Attempt at a Solution

Apart from knowing γ^'' should =0 I have no idea how to show that, I just need a bump in the right direction on how to start

Thanks for any help

 

[mfb]

Apart from knowing γ^'' should =0

That is not true. In general, acceleration won't be zero.

You can find the full trajectory, but you can also show that the magnitude of the velocity is constant without the trajectory. For mathematical reasons, it is easier to show that the squared velocity is constant, which is equivalent to the previous statement.