Velocity and acceleration vectors of charged particle moving perpendicular to the direction of magnetic field at a given instant of time are `vartheta= 2hat(i) + c hat(j) and bar(a) = 3 hat(i) + 4 hat(j)` respectively. Then the value of .c. is

Options

A. 3
B. 1.5
C. `-1.5`
D. `-3`

Answer

To solve the problem, we need to find the value of \( c \) given the velocity vector \( \vec{v} = 2\hat{i} + c\hat{j} \) and the acceleration vector \( \vec{a} = 3\hat{i} + 4\hat{j} \). Since the velocity of the charged particle is perpendicular to the magnetic field, the acceleration vector must also be perpendicular to the velocity vector.

### Step-by-step solution:

1. **Understand the condition of perpendicularity**:
Since the velocity vector \( \vec{v} \) and the acceleration vector \( \vec{a} \) are perpendicular, their dot product must equal zero:
\[ \vec{a} \cdot \vec{v} = 0 \]

2. **Write the dot product**:
Substitute the vectors into the dot product:
\[ (3\hat{i} + 4\hat{j}) \cdot (2\hat{i} + c\hat{j}) = 0 \]

3. **Calculate the dot product**:
Using the properties of dot product:
\[ 3 \cdot 2 + 4 \cdot c = 0 \]
This simplifies to:
\[ 6 + 4c = 0 \]

4. **Solve for \( c \)**:
Rearranging the equation gives:
\[ 4c = -6 \]
Dividing both sides by 4:
\[ c = -\frac{6}{4} = -1.5 \]

5. **Final answer**:
Therefore, the value of \( c \) is:
\[ c = -1.5 \]