Final answer:
To find the position vector, integrate the given acceleration function, apply the initial velocity condition for the constant of integration in the velocity function, and then integrate again while applying the initial position for the constant in the position function.
Step-by-step explanation:
The position vector of a particle can be obtained by integrating the acceleration function with respect to time and applying the initial conditions for velocity and position. Given the acceleration function a(t) = 11ti + e^t j + e^{-t} k, we integrate this function with respect to time to obtain the velocity function.
The integration will yield a velocity function that includes a constant of integration. This constant is determined by the initial velocity condition, which is given as v(0) = k.
Upon obtaining the velocity function, we perform another integration to find the position function, using the initial position r(0) = j to find the constant of integration for the position function. Therefore, the final position function will be determined by these integrations and application of initial conditions.