If `vec(F ) = (60 hat(i) + 15 hat(j) - 3 hat(k)) N` and `vec(V) = (2 hat(i) - 4 hat(j) + 5 hat(k))` m/s, then instantaneous power is:

Options

A. 195 watt
B. 45 watt
C. 75 watt
D. 100 watt

Explanation

To find the instantaneous power given the force and velocity vectors, we can use the formula for power, which is the dot product of the force vector \(\vec{F}\) and the velocity vector \(\vec{V}\).

Given:
\[ \vec{F} = 60 \hat{i} + 15 \hat{j} - 3 \hat{k} \quad \text{(in Newtons)} \]
\[ \vec{V} = 2 \hat{i} - 4 \hat{j} + 5 \hat{k} \quad \text{(in m/s)} \]

### Step 1: Calculate the dot product \(\vec{F} \cdot \vec{V}\)

The dot product of two vectors \(\vec{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\) and \(\vec{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}\) is given by:
\[ \vec{A} \cdot \vec{B} = a_1 b_1 + a_2 b_2 + a_3 b_3 \]

Applying this to our vectors:
\[ \vec{F} \cdot \vec{V} = (60)(2) + (15)(-4) + (-3)(5) \]

### Step 2: Perform the multiplication for each component

Calculating each term:
- For the \(\hat{i}\) components: \(60 \times 2 = 120\)
- For the \(\hat{j}\) components: \(15 \times -4 = -60\)
- For the \(\hat{k}\) components: \(-3 \times 5 = -15\)

### Step 3: Sum the results of the dot product

Now, we sum these results:
\[ \vec{F} \cdot \vec{V} = 120 - 60 - 15 \]

### Step 4: Calculate the final value

Calculating the sum:
\[ 120 - 60 = 60 \]
\[ 60 - 15 = 45 \]

### Conclusion

Thus, the instantaneous power is:
\[ P = 45 \text{ watts} \]

### Final Answer
The instantaneous power is \(45 \text{ watts}\).

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