A particle is moving in a curvillinear path defined by the equations `x=2t^2, y=t^2-4t and z=3t-5`.
Find out the magnitudes of the components of velocity and acceleration along `(hati-3hatj+2hatk)` at time t=1.

Explanation

Velocity , `vecv=(vec(dr))/(dt)=d/(dt)(xhati+yhatj+zhatk)`
`=(dx)/(dt)hati+(dy)/(dt)hatj+(dz)/(dt)hatk`
`=4thati+(2t-4)hatj+3hatk`
`therefore At" "t=1, vecv=4hati-2hatj+3hatk " ".....(1)`
Acceleration , `veca =(vec(dv))/(dt)=4hati+2hatj`= constant ...... (2)
Unit vector along `vecA=hati-3hatj+2hatk`
`hatn=(vecA)/(A)=(hati-3hatj+2hatk)/(sqrt(1^2+(-3)^2+2^2))=1/sqrt(14)(hati-3hatj+2hatk)" "....(2)`
`therefore` At t=1 , the component of `vecv` along `vecA` is , from (1) and (3),
`v_A=hatn*vecv=1/(sqrt(14))(hati-3hatj+2hatk)*(4hati-2hatj+3hatk)`
`=1/sqrt(14)xx16=(8sqrt(14))/(7)`
The component of `veca ` along `vecA` , from (2) and (3) ,
`a_A=hatn*veca=1/(sqrt(14))(hati-3hatj+2hatk)*(4hati+2hatj)`
`=1/(sqrt(14))xx(-2)=sqrt(14)/(7)`
`therefore |a_A|=sqrt(14)/(7)`.