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:

\[ P = \mathbf{F} \cdot \mathbf{v} \]

where \( P \) is the power, \( \mathbf{F} \) is the force vector, and \( \mathbf{v} \) is the velocity vector.

### Step 1: Write down the force and velocity vectors
Given:
\[ \mathbf{F} = 20 \hat{i} - 3 \hat{j} + 5 \hat{k} \, \text{N} \]
\[ \mathbf{v} = 6 \hat{i} + 20 \hat{j} - 3 \hat{k} \, \text{m/s} \]

### Step 2: Calculate the dot product \( \mathbf{F} \cdot \mathbf{v} \)
The dot product of two vectors \( \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \mathbf{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = a_1 b_1 + a_2 b_2 + a_3 b_3 \]

Applying this to our vectors:
\[ \mathbf{F} \cdot \mathbf{v} = (20)(6) + (-3)(20) + (5)(-3) \]

### Step 3: Perform the calculations
Calculating each term:
- \( 20 \times 6 = 120 \)
- \( -3 \times 20 = -60 \)
- \( 5 \times -3 = -15 \)

Now, sum these results:
\[ \mathbf{F} \cdot \mathbf{v} = 120 - 60 - 15 = 45 \]

### Step 4: Write the final answer for power
Thus, the power \( P \) is:
\[ P = 45 \, \text{W} \]

### Final Answer
The power of the engine is \( 45 \, \text{W} \).

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