The motion of a particle is defined by the position vector
`vec(r)=(cost)hat(i)+(sin t)hat(j)` Where `t` is expressed in seconds. Determine the value of `t` for which positions vectors and velocity vector are perpendicular.

Options

A. `1` sec
B. `2` sec
C. for all value of `t`
D. None of these

Explanation

To determine the value of \( t \) for which the position vector \( \vec{r} \) and the velocity vector \( \vec{v} \) are perpendicular, we can follow these steps:

### Step 1: Define the Position Vector
The position vector is given as:
\[ \vec{r}(t) = \cos(t) \hat{i} + \sin(t) \hat{j} \]

### Step 2: Calculate the Velocity Vector
The velocity vector \( \vec{v}(t) \) is the derivative of the position vector with respect to time \( t \):
\[ \vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{d}{dt}[\cos(t) \hat{i} + \sin(t) \hat{j}] \]
Using the derivatives of sine and cosine:
\[ \frac{d}{dt}[\cos(t)] = -\sin(t) \quad \text{and} \quad \frac{d}{dt}[\sin(t)] = \cos(t) \]
Thus, the velocity vector becomes:
\[ \vec{v}(t) = -\sin(t) \hat{i} + \cos(t) \hat{j} \]

### Step 3: Check for Perpendicularity
Two vectors are perpendicular if their dot product is zero:
\[ \vec{r}(t) \cdot \vec{v}(t) = 0 \]
Calculating the dot product:
\[ \vec{r}(t) \cdot \vec{v}(t) = (\cos(t) \hat{i} + \sin(t) \hat{j}) \cdot (-\sin(t) \hat{i} + \cos(t) \hat{j}) \]
Expanding this:
\[ = \cos(t)(-\sin(t)) + \sin(t)(\cos(t)) \]
\[ = -\cos(t)\sin(t) + \sin(t)\cos(t) = 0 \]

### Step 4: Conclusion
Since the dot product is zero for all values of \( t \), the position vector and the velocity vector are perpendicular for all values of \( t \).

### Final Answer
The value of \( t \) for which the position vector and the velocity vector are perpendicular is:
\[ \text{For all values of } t \]

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