A proton initially has `vec v = 4.0 hati - 2.0 hatj + 3.0 hatk` and then 4.0 s later has `vec v = -2.0 hati - 2.0 hatj + 5.0hatk ` (in meter per second) for that 4.0s what are (a) the proton's average acceleration `vec a_(avg) ` in unit vector notation (b) the magnitude of `vec a_(avg)` and (c ) the angle between `veca_(avg)` and the positive direction of the x axis ?

Explanation

We use eq. with vec v_1` designating the initial velocity and `vec v_2` designating the later one.
(a) The average acceleration during the `Delta t = 4s` interval is
`vec a_(avg) = ((-2.0 hati - 2.0 hatj + 5.0 hatk) m//s - (4.0hati - 22hatj + 3.0 hatk)m//s)/(4s)`
` = (-1.5 m//s^2) hati + (0.5 m//s^2) hatk`
(b) The magnitude of `vec a_(avg) ` is
` sqrt( (-1.5 m//s^2)^2 + (0.5m//s^2)^2) = 1.6 m//s^2`
( c) Its angle in the xz plane (measured from the +x axis is one of these possibilities :
` tan^(-1) ( (0.5m//s^2)/(-1.5 m//s^2)) = - 18^@ or 162^@`