A particle moves in XY plane such that its position, velocity and acceleration are given by
`vec(r)=xhat(i)+yhat(j), " "vec(v)=v_(x)hat(i)+v_(y)hat(j), " "vec(a)=a_(x)hat(i)+a_(y)hat(j)`
Which of the following condition is correct if the particle is speeding down?

A. `xv_(x)+yv_(y) lt 0`
B. `xv_(x)+yv_(y) gt 0`
C. `a_(x)v_(x)+a_(y)v_(y) lt 0`
D. `a_(x)v_(x)+a_(y)v_(y) gt 0`

Options

A. `xv_(x)+yv_(y) lt 0`
B. `xv_(x)+yv_(y) gt 0`
C. `a_(x)v_(x)+a_(y)v_(y) lt 0`
D. `a_(x)v_(x)+a_(y)v_(y) gt 0`

Explanation

To solve the problem, we need to analyze the conditions under which a particle is "speeding down," which means that its speed is decreasing.

### Step-by-Step Solution:

1. **Understanding Speeding Down**:
- A particle is said to be speeding down when its velocity is decreasing. This can happen when the acceleration vector is acting in the opposite direction to the velocity vector.

2. **Vectors Involved**:
- The position vector is given as \( \vec{r} = x \hat{i} + y \hat{j} \).
- The velocity vector is \( \vec{v} = v_x \hat{i} + v_y \hat{j} \).
- The acceleration vector is \( \vec{a} = a_x \hat{i} + a_y \hat{j} \).

3. **Dot Product of Acceleration and Velocity**:
- To determine if the particle is speeding down, we can use the dot product of the acceleration vector and the velocity vector:
\[ \vec{a} \cdot \vec{v} = a_x v_x + a_y v_y \]
- If this dot product is negative, it indicates that the acceleration is in the opposite direction to the velocity, which means the particle is slowing down.

4. **Condition for Speeding Down**:
- Therefore, for the particle to be speeding down, we require:
\[ a_x v_x + a_y v_y < 0 \]

5. **Analyzing the Options**:
- Now, we check the given options:
- A. \( x v_x + y v_y < 0 \) (Not relevant to our condition)
- B. \( x v_x + y v_y > 0 \) (Not relevant to our condition)
- C. \( a_x v_x + a_y v_y < 0 \) (This matches our derived condition)
- D. \( a_x v_x + a_y v_y > 0 \) (This contradicts our derived condition)

6. **Conclusion**:
- The correct condition for the particle to be speeding down is given by option C:
\[ a_x v_x + a_y v_y < 0 \]

### Final Answer:
**C. \( a_x v_x + a_y v_y < 0 \)**