Velocity and acceleration vector of a charged particle moving in a magnetic field at some instant are `vecv=3hati+4hatj and veca=2hati+xhatj`. Select the correct options.

Options

A. `x=-1.5`
B. `x=3`
C. Magnetic field is along z-direction
D. Kinetic energy of the particle is constant

Answer

To solve the problem, we need to analyze the given vectors for velocity and acceleration of a charged particle moving in a magnetic field. The velocity vector is given as \(\vec{v} = 3\hat{i} + 4\hat{j}\) and the acceleration vector is given as \(\vec{a} = 2\hat{i} + x\hat{j}\).

### Step-by-Step Solution:

1. **Understanding the Relationship Between Velocity and Acceleration**:
- In a magnetic field, the magnetic force acting on a charged particle is given by \(\vec{F} = q(\vec{v} \times \vec{B})\).
- This force is always perpendicular to the velocity vector \(\vec{v}\). Therefore, the acceleration vector \(\vec{a}\) will also be perpendicular to \(\vec{v}\).

2. **Using the Dot Product to Find \(x\)**:
- Since \(\vec{v}\) and \(\vec{a}\) are perpendicular, their dot product must equal zero:
\[ \vec{v} \cdot \vec{a} = 0 \]
- Calculate the dot product:
\[ (3\hat{i} + 4\hat{j}) \cdot (2\hat{i} + x\hat{j}) = 3 \cdot 2 + 4 \cdot x = 6 + 4x \]
- Set the dot product equal to zero:
\[ 6 + 4x = 0 \]

3. **Solving for \(x\)**:
- Rearranging the equation gives:
\[ 4x = -6 \implies x = -\frac{6}{4} = -1.5 \]

4. **Conclusion About the Options**:
- From the calculation, we find \(x = -1.5\). Thus, the correct option related to the value of \(x\) is confirmed.
- The kinetic energy of the particle remains constant because the magnetic force does no work (it is always perpendicular to the displacement).
- The magnetic field direction can be determined from the cross product relationship, but it is not necessarily along the z-axis.

### Final Answer:
- The value of \(x\) is \(-1.5\).
- The correct options based on the analysis are A and D.