Answer:
![r(t)=](https://tex.z-dn.net/?f=r%28t%29%3D%3C%5Cfrac%7B7%7D%7B2%7Dt%5E2%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7Dt%2B2%2C%5Cfrac%7B7%7D%7B2%7Dt%5E2%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7Dt%2B1%2C-%5Cfrac%7B1%7D%7B2%7Dt%5E2-%5Cfrac%7B8%7D%7B3%5Csqrt%7B11%7D%7Dt%2B6%3E)
Step-by-step explanation:
We are given that
Initial position of particle =![r_0=r(0)=](https://tex.z-dn.net/?f=r_0%3Dr%280%29%3D%3C2%2C1%2C6%3E)
Initial speed of the particle = 8 m/s
Acceleration=7 i+7j-k=<7,7,-1>
We have to find the equation of position vector r(t) of the particle at time t.
We know that
Acceleration=a=![\frac{dv}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7Bdv%7D%7Bdt%7D)
![dv=adt](https://tex.z-dn.net/?f=dv%3Dadt)
![dv=(7i+7j-k)dt](https://tex.z-dn.net/?f=dv%3D%287i%2B7j-k%29dt)
Integrating on both sides then, we get
![v(t)=+v_0](https://tex.z-dn.net/?f=v%28t%29%3D%3C7t%2C7t-t%3E%2Bv_0)
Where ![v_0=,v_1,v_2,v_3>](https://tex.z-dn.net/?f=v_0%3D%2Cv_1%2Cv_2%2Cv_3%3E)
![v_0=v(0)=8](https://tex.z-dn.net/?f=v_0%3Dv%280%29%3D8)
![\mid \mid=8](https://tex.z-dn.net/?f=%5Cmid%20%3Cv_1%2Cv_2%2Cv_3%3E%5Cmid%3D8)
Since, the particle travel in straight line to (9,8,5) and (9,8,5)-(2,1,6)=<7,7,-1>
There exist a constant s such that
![v_1=7s, v_2=7s, v_3=-s](https://tex.z-dn.net/?f=v_1%3D7s%2C%20v_2%3D7s%2C%20v_3%3D-s)
![\sqrt{v^2_1+v^2_2+v^2_3}=8](https://tex.z-dn.net/?f=%5Csqrt%7Bv%5E2_1%2Bv%5E2_2%2Bv%5E2_3%7D%3D8)
![\sqrt{49s^2+49s^2+s^2}=8](https://tex.z-dn.net/?f=%5Csqrt%7B49s%5E2%2B49s%5E2%2Bs%5E2%7D%3D8)
![\sqrt{99s^2}=8](https://tex.z-dn.net/?f=%5Csqrt%7B99s%5E2%7D%3D8)
![99s^2=64](https://tex.z-dn.net/?f=99s%5E2%3D64)
![s^2=\frac{64}{99}](https://tex.z-dn.net/?f=s%5E2%3D%5Cfrac%7B64%7D%7B99%7D)
![s=\sqrt{\frac{64}{99}}=\frac{8}{3\sqrt{11}}](https://tex.z-dn.net/?f=s%3D%5Csqrt%7B%5Cfrac%7B64%7D%7B99%7D%7D%3D%5Cfrac%7B8%7D%7B3%5Csqrt%7B11%7D%7D)
![v_0=](https://tex.z-dn.net/?f=v_0%3D%3C%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7D%2C%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7D%2C-%5Cfrac%7B8%7D%7B3%5Csqrt%7B11%7D%7D%3E)
![v(t)=+](https://tex.z-dn.net/?f=v%28t%29%3D%3C7t%2C7t%2C-t%3E%2B%3C%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7D%2C%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7D%2C-%5Cfrac%7B8%7D%7B3%5Csqrt%7B11%7D%7D%3E)
![v(t)=](https://tex.z-dn.net/?f=v%28t%29%3D%3C7t%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7D%2C7t%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7D%2C-t-%5Cfrac%7B8%7D%7B3%5Csqrt%7B11%7D%7D%3E)
![dr=vdt](https://tex.z-dn.net/?f=dr%3Dvdt)
Integrating on both sides then we get
![r(t)=\int dt](https://tex.z-dn.net/?f=r%28t%29%3D%5Cint%20%3C7t%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7D%2C7t%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7D%2C-t-%5Cfrac%7B8%7D%7B3%5Csqrt%7B11%7D%7D%3E%20dt)
![r(t)= +r_0](https://tex.z-dn.net/?f=r%28t%29%3D%20%3C%5Cfrac%7B7%7D%7B2%7Dt%5E2%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7Dt%2C%5Cfrac%7B7%7D%7B2%7Dt%5E2%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7Dt%2C-%5Cfrac%7Bt%5E2%7D%7B2%7D-%5Cfrac%7B8%7D%7B3%5Csqrt%7B11%7D%7Dt%3E%2Br_0)
Where
=Some constant vector
![r_0=](https://tex.z-dn.net/?f=r_0%3D%3C2%2C1%2C6%3E)
Substitute the value
![r(t)=+](https://tex.z-dn.net/?f=r%28t%29%3D%3C%5Cfrac%7B7%7D%7B2%7Dt%5E2%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7Dt%2C%5Cfrac%7B7%7D%7B2%7Dt%5E2%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7Dt%2C-%5Cfrac%7B1%7D%7B2%7Dt%5E2-%5Cfrac%7B8%7D%7B3%5Csqrt%7B11%7D%7Dt%3E%2B%3C2%2C1%2C6%3E)
![r(t)=](https://tex.z-dn.net/?f=r%28t%29%3D%3C%5Cfrac%7B7%7D%7B2%7Dt%5E2%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7Dt%2B2%2C%5Cfrac%7B7%7D%7B2%7Dt%5E2%2B%5Cfrac%7B56%7D%7B3%5Csqrt%7B11%7D%7Dt%2B1%2C-%5Cfrac%7B1%7D%7B2%7Dt%5E2-%5Cfrac%7B8%7D%7B3%5Csqrt%7B11%7D%7Dt%2B6%3E)
This is required equation of the position vector r(t) of the particle at time t.