The position vector of a particle is `r = a sin omega t hati +a cos omega t hatj`
The velocity of the particle is

Options

A. parallel to position vector
B. perpendicular to position vector
C. directed towards origin
D. directed away from the origin

Explanation

To find the velocity of the particle given its position vector \( \mathbf{r} = a \sin(\omega t) \hat{i} + a \cos(\omega t) \hat{j} \), we will follow these steps:

### Step 1: Differentiate the Position Vector
The velocity vector \( \mathbf{v} \) is the time derivative of the position vector \( \mathbf{r} \).

\[ \mathbf{v} = \frac{d\mathbf{r}}{dt} \]

### Step 2: Apply the Chain Rule
Differentiate each component of \( \mathbf{r} \):

1. For the \( \hat{i} \) component:
\[ \frac{d}{dt}(a \sin(\omega t)) = a \omega \cos(\omega t) \]

2. For the \( \hat{j} \) component:
\[ \frac{d}{dt}(a \cos(\omega t)) = -a \omega \sin(\omega t) \]

### Step 3: Combine the Results
Now, combine the derivatives to form the velocity vector:

\[ \mathbf{v} = a \omega \cos(\omega t) \hat{i} - a \omega \sin(\omega t) \hat{j} \]

### Step 4: Analyze the Relationship Between Position and Velocity Vectors
To find the relationship between the position vector \( \mathbf{r} \) and the velocity vector \( \mathbf{v} \), we can take the dot product \( \mathbf{r} \cdot \mathbf{v} \):

\[ \mathbf{r} \cdot \mathbf{v} = (a \sin(\omega t) \hat{i} + a \cos(\omega t) \hat{j}) \cdot (a \omega \cos(\omega t) \hat{i} - a \omega \sin(\omega t) \hat{j}) \]

### Step 5: Calculate the Dot Product
Calculating the dot product:

\[ \mathbf{r} \cdot \mathbf{v} = a^2 \sin(\omega t) \cdot a \omega \cos(\omega t) - a^2 \cos(\omega t) \cdot a \omega \sin(\omega t) \]

This simplifies to:

\[ \mathbf{r} \cdot \mathbf{v} = a^2 \omega \sin(\omega t) \cos(\omega t) - a^2 \omega \cos(\omega t) \sin(\omega t) = 0 \]

### Step 6: Conclusion
Since the dot product \( \mathbf{r} \cdot \mathbf{v} = 0 \), this implies that the position vector \( \mathbf{r} \) and the velocity vector \( \mathbf{v} \) are perpendicular to each other.

Thus, the answer is that the velocity of the particle is **perpendicular to the position vector**.

### Final Answer
The velocity of the particle is perpendicular to the position vector.

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