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

Options

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

Answer

To find the instantaneous power given the force vector \(\vec{F}\) and the velocity vector \(\vec{v}\), we can use the formula for instantaneous power:

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

where \(P\) is the instantaneous power, \(\vec{F}\) is the force vector, and \(\vec{v}\) is the velocity vector. The dot product of two vectors is calculated as follows:

\[ \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z \]

### Step-by-step Solution:

1. **Identify the vectors:**
- Force vector: \(\vec{F} = 60 \hat{i} + 15 \hat{j} - 3 \hat{k}\)
- Velocity vector: \(\vec{v} = 2 \hat{i} - 4 \hat{j} + 5 \hat{k}\)

2. **Calculate the dot product:**
- The dot product \(\vec{F} \cdot \vec{v}\) is calculated as follows:
\[ P = (60 \hat{i} + 15 \hat{j} - 3 \hat{k}) \cdot (2 \hat{i} - 4 \hat{j} + 5 \hat{k}) \]

3. **Compute each component:**
- For the \(\hat{i}\) components:
\[ 60 \cdot 2 = 120 \]
- For the \(\hat{j}\) components:
\[ 15 \cdot (-4) = -60 \]
- For the \(\hat{k}\) components:
\[ -3 \cdot 5 = -15 \]

4. **Combine the results:**
- Now, sum the results of the dot product:
\[ P = 120 - 60 - 15 \]
\[ P = 120 - 60 = 60 \]
\[ P = 60 - 15 = 45 \]

5. **Final answer:**
- Therefore, the instantaneous power is:
\[ P = 45 \text{ watts} \]