A body is undergoing circular motion. Some vector expression that describe the motion are given below
(a) `vecV=vecomega xx vecR`
(b) `a_(vecT)=vec alpha xx vecR`
(c ) `a_(vecC)=vecalphaxx vecR`
Of the above statements following are/is correct

Options

A. a, b and c
B. a and b
C. b and c
D. c and a

Answer

To solve the problem, we need to analyze the given vector expressions related to circular motion and determine which of them are correct. Let's break down the statements one by one.

### Step 1: Analyze Statement (a)
The first statement is:
\[ \vec{V} = \vec{\omega} \times \vec{R} \]
Where:
- \(\vec{V}\) is the linear velocity vector.
- \(\vec{\omega}\) is the angular velocity vector.
- \(\vec{R}\) is the radial vector (position vector from the center of the circular path to the particle).

In circular motion, the linear velocity \(\vec{V}\) of a particle is indeed given by the cross product of the angular velocity \(\vec{\omega}\) and the radial vector \(\vec{R}\). This statement is correct.

### Step 2: Analyze Statement (b)
The second statement is:
\[ \vec{a}_{\text{T}} = \vec{\alpha} \times \vec{R} \]
Where:
- \(\vec{a}_{\text{T}}\) is the tangential acceleration vector.
- \(\vec{\alpha}\) is the angular acceleration vector.

Tangential acceleration is related to the change in angular velocity over time. The correct expression for tangential acceleration is given by:
\[ \vec{a}_{\text{T}} = \alpha \cdot \vec{R} \]
This means that the tangential acceleration is directly proportional to the angular acceleration and not through a cross product. Therefore, this statement is incorrect.

### Step 3: Analyze Statement (c)
The third statement is:
\[ \vec{a}_{\text{C}} = \vec{\alpha} \times \vec{R} \]
Where:
- \(\vec{a}_{\text{C}}\) is the centripetal acceleration vector.

Centripetal acceleration is directed towards the center of the circular path and is given by:
\[ \vec{a}_{\text{C}} = \frac{V^2}{R} \text{ (or equivalently, } \frac{\omega^2 R}{R} = \omega^2 R\text{)} \]
It is not expressed as a cross product with angular acceleration. Therefore, this statement is also incorrect.

### Conclusion
From the analysis:
- Statement (a) is **correct**.
- Statement (b) is **incorrect**.
- Statement (c) is **incorrect**.

Thus, the correct statements are:
- Only statement (a) is correct.

### Final Answer
The correct answer is: **Only statement (a) is correct.**