The position vector of a particle is given by `vec(r)=1.2 t hat(i)+0.9 t^(2) hat(j)-0.6(t^(3)-1)hat(k)` where t is the time in seconds from the start of motion and where `vec(r)` is expressed in meters. For the condition when `t=4` second, determine the power `(P=vec(F).vec(v))` in watts produced by the force `vec(F)=(60hat(i)-25hat(j)-40hat(k)) N` which is acting on the particle.

Explanation

To solve the problem, we need to follow these steps:

### Step 1: Write down the position vector
The position vector of the particle is given by:
\[ \vec{r} = 1.2 t \hat{i} + 0.9 t^2 \hat{j} - 0.6(t^3 - 1) \hat{k} \]

### Step 2: Calculate the velocity vector
The velocity vector \(\vec{v}\) is the derivative of the position vector \(\vec{r}\) with respect to time \(t\):
\[ \vec{v} = \frac{d\vec{r}}{dt} = \frac{d}{dt}(1.2 t \hat{i} + 0.9 t^2 \hat{j} - 0.6(t^3 - 1) \hat{k}) \]
Calculating the derivatives:
- For \(1.2 t \hat{i}\), the derivative is \(1.2 \hat{i}\).
- For \(0.9 t^2 \hat{j}\), the derivative is \(1.8 t \hat{j}\).
- For \(-0.6(t^3 - 1) \hat{k}\), the derivative is \(-1.8 t^2 \hat{k}\).

Thus, the velocity vector is:
\[ \vec{v} = 1.2 \hat{i} + 1.8 t \hat{j} - 1.8 t^2 \hat{k} \]

### Step 3: Evaluate the velocity at \(t = 4\) seconds
Substituting \(t = 4\) into the velocity equation:
\[ \vec{v} = 1.2 \hat{i} + 1.8(4) \hat{j} - 1.8(4^2) \hat{k} \]
Calculating the components:
- \(1.8(4) = 7.2\)
- \(1.8(4^2) = 1.8(16) = 28.8\)

Thus,
\[ \vec{v} = 1.2 \hat{i} + 7.2 \hat{j} - 28.8 \hat{k} \]

### Step 4: Write down the force vector
The force vector is given as:
\[ \vec{F} = 60 \hat{i} - 25 \hat{j} - 40 \hat{k} \]

### Step 5: Calculate the dot product \(\vec{F} \cdot \vec{v}\)
The power \(P\) is given by the dot product of the force and velocity vectors:
\[ P = \vec{F} \cdot \vec{v} = (60 \hat{i} - 25 \hat{j} - 40 \hat{k}) \cdot (1.2 \hat{i} + 7.2 \hat{j} - 28.8 \hat{k}) \]
Calculating the dot product:
\[ P = 60 \cdot 1.2 + (-25) \cdot 7.2 + (-40) \cdot (-28.8) \]
Calculating each term:
- \(60 \cdot 1.2 = 72\)
- \(-25 \cdot 7.2 = -180\)
- \(-40 \cdot -28.8 = 1152\)

Thus,
\[ P = 72 - 180 + 1152 = 1044 \text{ watts} \]

### Final Result
The power produced by the force acting on the particle at \(t = 4\) seconds is:
\[ P = 1044 \text{ watts} \]