A ice block floating in a river is pushed through a displacement `vec(d)=(20m) hat(i) - (16 m) hat(j)` along a straight embankment by rushing water, which exerts a force `vec(F)= (210 N) hat(i)-(150 N) hat(j)` on the block. How much work does the force do on the block during the displacement ?

Answer

To find the work done by the force on the ice block during the displacement, we will use the formula for work done, which is given by the dot product of the force vector and the displacement vector.

### Step-by-Step Solution:

1. **Identify the Given Vectors**:
- Displacement vector: \(\vec{d} = 20 \hat{i} - 16 \hat{j}\) (in meters)
- Force vector: \(\vec{F} = 210 \hat{i} - 150 \hat{j}\) (in Newtons)

2. **Calculate the Dot Product**:
The work done \(W\) is calculated using the formula:
\[ W = \vec{F} \cdot \vec{d} \]
The dot product of two vectors \(\vec{A} = a_1 \hat{i} + a_2 \hat{j}\) and \(\vec{B} = b_1 \hat{i} + b_2 \hat{j}\) is given by:
\[ \vec{A} \cdot \vec{B} = a_1 b_1 + a_2 b_2 \]

Applying this to our vectors:
\[ W = (210 \hat{i} - 150 \hat{j}) \cdot (20 \hat{i} - 16 \hat{j}) \]
\[ W = (210 \times 20) + (-150 \times -16) \]

3. **Perform the Multiplications**:
- Calculate \(210 \times 20\):
\[ 210 \times 20 = 4200 \]
- Calculate \(-150 \times -16\):
\[ -150 \times -16 = 2400 \]

4. **Add the Results**:
Now, add the two results together:
\[ W = 4200 + 2400 = 6600 \text{ Joules} \]

5. **Final Answer**:
The work done by the force on the block during the displacement is:
\[ \boxed{6600 \text{ Joules}} \]