A particle travels so that its acceleration is given by `vec(a)=5 cos t hat(i)-3 sin t hat(j)`. If the particle is located at `(-3, 2)` at time `t=0` and is moving with a velocity given by `(-3hat(i)+2hat(j))`. Find
(i) The velocity `[vec(v)=int vec(a).dt]` at time t and
(ii) The position vector `[vec(r)=int vec(v).dt]` of the particle at time `t (t gt 0)`.

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the velocity vector \( \vec{v}(t) \)
Given the acceleration vector:
\[ \vec{a} = 5 \cos(t) \hat{i} - 3 \sin(t) \hat{j} \]
We know that the velocity vector can be found by integrating the acceleration vector with respect to time \( t \):
\[ \vec{v}(t) = \int \vec{a} \, dt + \vec{v}_0 \]
where \( \vec{v}_0 \) is the initial velocity vector.

The initial velocity vector is given as:
\[ \vec{v}_0 = -3 \hat{i} + 2 \hat{j} \]

Now we can integrate the components of the acceleration vector:
1. For the \( \hat{i} \) component:
\[ v_x(t) = \int 5 \cos(t) \, dt = 5 \sin(t) + C_1 \]
2. For the \( \hat{j} \) component:
\[ v_y(t) = \int -3 \sin(t) \, dt = 3 \cos(t) + C_2 \]

Now we need to determine the constants \( C_1 \) and \( C_2 \) using the initial conditions at \( t = 0 \):
- At \( t = 0 \):
\[ v_x(0) = -3 \Rightarrow 5 \sin(0) + C_1 = -3 \Rightarrow C_1 = -3 \]
\[ v_y(0) = 2 \Rightarrow 3 \cos(0) + C_2 = 2 \Rightarrow C_2 = -1 \]

Thus, the velocity vector becomes:
\[ \vec{v}(t) = (5 \sin(t) - 3) \hat{i} + (3 \cos(t) - 1) \hat{j} \]

### Step 2: Find the position vector \( \vec{r}(t) \)
Next, we find the position vector by integrating the velocity vector:
\[ \vec{r}(t) = \int \vec{v}(t) \, dt + \vec{r}_0 \]
where \( \vec{r}_0 \) is the initial position vector.

The initial position vector is given as:
\[ \vec{r}_0 = -3 \hat{i} + 2 \hat{j} \]

Now, we can integrate the components of the velocity vector:
1. For the \( \hat{i} \) component:
\[ x(t) = \int (5 \sin(t) - 3) \, dt = -5 \cos(t) - 3t + C_3 \]
2. For the \( \hat{j} \) component:
\[ y(t) = \int (3 \cos(t) - 1) \, dt = 3 \sin(t) - t + C_4 \]

Again, we determine the constants \( C_3 \) and \( C_4 \) using the initial conditions at \( t = 0 \):
- At \( t = 0 \):
\[ x(0) = -3 \Rightarrow -5 \cos(0) - 3(0) + C_3 = -3 \Rightarrow C_3 = 2 \]
\[ y(0) = 2 \Rightarrow 3 \sin(0) - 0 + C_4 = 2 \Rightarrow C_4 = 2 \]

Thus, the position vector becomes:
\[ \vec{r}(t) = (-5 \cos(t) - 3t + 2) \hat{i} + (3 \sin(t) - t + 2) \hat{j} \]

### Final Results:
1. The velocity vector at time \( t \):
\[ \vec{v}(t) = (5 \sin(t) - 3) \hat{i} + (3 \cos(t) - 1) \hat{j} \]
2. The position vector at time \( t \):
\[ \vec{r}(t) = (-5 \cos(t) - 3t + 2) \hat{i} + (3 \sin(t) - t + 2) \hat{j} \]