An object is moving in the x-y plane with the position as a function of time given by `vecr = x(t)hati + y(t)hatj`. Point O is at `x = 0 , y = 0` . The object is definitely moving towards O when

Options

A. `v_xgt0,v_ygt0`
B. `v_xlt0,v_ylt0`
C. `xv_x+yv_ylt0`
D. `xv_x+yv_ygt0`

Explanation

To determine when an object moving in the x-y plane is definitely moving towards the origin (point O at coordinates (0,0)), we can analyze the relationship between the position vector and the velocity vector of the object.

### Step-by-Step Solution:

1. **Define the Position Vector**:
The position vector of the object is given by:
\[ \vec{r} = x(t) \hat{i} + y(t) \hat{j} \]
where \(x(t)\) and \(y(t)\) are the x and y coordinates as functions of time.

2. **Define the Velocity Vector**:
The velocity vector \(\vec{v}\) of the object can be derived by differentiating the position vector with respect to time:
\[ \vec{v} = \frac{d\vec{r}}{dt} = \frac{dx}{dt} \hat{i} + \frac{dy}{dt} \hat{j} = v_x \hat{i} + v_y \hat{j} \]
where \(v_x = \frac{dx}{dt}\) and \(v_y = \frac{dy}{dt}\).

3. **Condition for Moving Towards the Origin**:
For the object to be moving towards the origin, the velocity vector must have a component that points in the direction of the negative position vector. This can be expressed mathematically using the dot product:
\[ \vec{r} \cdot \vec{v} < 0 \]
This means that the angle between the position vector and the velocity vector is greater than 90 degrees.

4. **Calculate the Dot Product**:
The dot product of the position vector and the velocity vector is:
\[ \vec{r} \cdot \vec{v} = (x(t) \hat{i} + y(t) \hat{j}) \cdot (v_x \hat{i} + v_y \hat{j}) = x(t)v_x + y(t)v_y \]

5. **Set Up the Inequality**:
To satisfy the condition for moving towards the origin, we set up the inequality:
\[ x(t)v_x + y(t)v_y < 0 \]

6. **Conclusion**:
The object is definitely moving towards the origin when the expression \(x(t)v_x + y(t)v_y\) is less than zero.

### Final Answer:
The object is moving towards point O (0,0) when:
\[ x(t)v_x + y(t)v_y < 0 \]