An engine exerts a force `F=(20hati-3hatj+5hatk)` N and moves with velocity `v=(6hati+20hatj-3hatk) ms^(-1)`. The power of the engine (in watt) is

Options

A. 45
B. 75
C. 20
D. 10

Explanation

To find the power of the engine, we can use the formula for power in terms of force and velocity:

### Step 1: Write down the formula for power
The power \( P \) is given by the dot product of force \( \vec{F} \) and velocity \( \vec{v} \):
\[ P = \vec{F} \cdot \vec{v} \]

### Step 2: Substitute the given values for force and velocity
We have:
\[ \vec{F} = 20 \hat{i} - 3 \hat{j} + 5 \hat{k} \, \text{N} \]
\[ \vec{v} = 6 \hat{i} + 20 \hat{j} - 3 \hat{k} \, \text{m/s} \]
Substituting these into the power formula gives:
\[ P = (20 \hat{i} - 3 \hat{j} + 5 \hat{k}) \cdot (6 \hat{i} + 20 \hat{j} - 3 \hat{k}) \]

### Step 3: Calculate the dot product
Using the properties of the dot product, we calculate:
\[ P = (20 \cdot 6) + (-3 \cdot 20) + (5 \cdot -3) \]
Calculating each term:
- \( 20 \cdot 6 = 120 \)
- \( -3 \cdot 20 = -60 \)
- \( 5 \cdot -3 = -15 \)

Now, combine these results:
\[ P = 120 - 60 - 15 \]

### Step 4: Simplify the expression
Now, simplifying the expression:
\[ P = 120 - 60 - 15 = 120 - 75 = 45 \, \text{W} \]

### Final Answer
Thus, the power of the engine is:
\[ \boxed{45 \, \text{W}} \]

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