The dimensions of the quantity `vecE xx vecB`, where `vecE` represents the electric field and `vecB` the magnetic field may be given as :-

Options

A. `[MT^(-3)]`
B. `[M^(2)LT^(-5)A^(-2)]`
C. `[M^(2)LT^(-3)A^(-1)]`
D. `[MLT^(-2)A^(-2)]`

Explanation

To find the dimensions of the quantity \( \vec{E} \times \vec{B} \), where \( \vec{E} \) represents the electric field and \( \vec{B} \) represents the magnetic field, we can follow these steps:

### Step 1: Determine the dimensions of the electric field \( \vec{E} \)
The electric field \( \vec{E} \) can be defined as the force \( \vec{F} \) per unit charge \( q \):
\[ \vec{E} = \frac{\vec{F}}{q} \]
The dimensions of force \( \vec{F} \) are given by:
\[ \text{Force} = \text{mass} \times \text{acceleration} = M \cdot L \cdot T^{-2} \]
The dimension of charge \( q \) is:
\[ \text{Charge} = I \cdot T = A \cdot T \]
Thus, the dimensions of the electric field \( \vec{E} \) can be calculated as:
\[ [\vec{E}] = \frac{[F]}{[q]} = \frac{M \cdot L \cdot T^{-2}}{A \cdot T} = M \cdot L \cdot A^{-1} \cdot T^{-1} \]

### Step 2: Determine the dimensions of the magnetic field \( \vec{B} \)
The magnetic field \( \vec{B} \) can also be defined in terms of force per unit charge and velocity:
\[ \vec{B} = \frac{\vec{F}}{q \cdot v} \]
Where \( v \) is the velocity. The dimensions of velocity \( v \) are:
\[ [v] = \frac{L}{T} \]
Thus, the dimensions of the magnetic field \( \vec{B} \) can be calculated as:
\[ [\vec{B}] = \frac{[F]}{[q] \cdot [v]} = \frac{M \cdot L \cdot T^{-2}}{A \cdot T \cdot L \cdot T^{-1}} = M \cdot A^{-1} \cdot T^{-2} \]

### Step 3: Calculate the dimensions of \( \vec{E} \times \vec{B} \)
Now, we can find the dimensions of the cross product \( \vec{E} \times \vec{B} \):
\[ [\vec{E} \times \vec{B}] = [\vec{E}] \cdot [\vec{B}] = (M \cdot L \cdot A^{-1} \cdot T^{-1}) \cdot (M \cdot A^{-1} \cdot T^{-2}) \]
Calculating this gives:
\[ [\vec{E} \times \vec{B}] = M^2 \cdot L \cdot A^{-2} \cdot T^{-3} \]

### Step 4: Finalize the dimensions
The final dimensions of the quantity \( \vec{E} \times \vec{B} \) are:
\[ [\vec{E} \times \vec{B}] = M^2 \cdot L \cdot T^{-3} \cdot A^{-2} \]

### Conclusion
Thus, the dimensions of the quantity \( \vec{E} \times \vec{B} \) can be expressed as:
\[ M^2 \cdot L \cdot T^{-3} \cdot A^{-2} \]