Final answer:
To find the position vector r(t), we integrate each component of the velocity vector r'(t) separately, and then apply the initial condition r(0) = i + j + k to determine the constants of integration. The resulting position vector is given by r(t) = ((1/7)t^7 + 1)i + (e^t)j + (e^(4t) + 1)k.
Step-by-step explanation:
The student is looking to find the position vector r(t) given that the derivative of r(t), or the velocity vector r'(t), is t⁴ i + et j + 4te4t k and the initial position vector r(0) is given as i + j + k. To find the position vector, we integrate each component of the velocity vector r'(t) with respect to time, and then apply the initial conditions to solve for the constants of integration. As provided by the student, the general format for expressing a position vector in terms of its components is r(t) = x(t)î + y(t)ĵ + z(t)k.
For the i component, we integrate t⁴ with respect to t to obtain (1/7)t⁹ î. For the j component, we integrate et, which is simply et ĵ, and for the k component, we integrate 4te4t, which yields (e4t &kcirc;) minus a constant that we will determine from the initial condition. Adding the constants of integration C₁, C₂, C₃ to each component and using the provided initial condition r(0) = i + j + k, allows us to solve for these constants.
Integrating each component:
- i component: ∫t⁴dt = (1/7)t⁹ + C₁
- j component: ∫etdt = et + C₂
- k component: ∫4te4tdt = e4t + C₃
Applying the initial condition:
- When t = 0, (1/7)(0)î + e0 ĵ + (0)&kcirc; + (C1î + C2ĵ + C3&kcirc;) must equal i + j + k.
- Therefore, C₁ = 1, C₂ = 0, and C₃ = 1.
So, the position vector r(t) is:
r(t) = ((1/7)t⁹ + 1)î + (et + 0)ĵ + (e4t + 1)&kcirc;.