Given: `vec p = - hati -3 hatj + 2hatk` and `vec r = hati + 3 hatj + 5hatk`. Find vector parallel to electric field at position r. [ Note that `vec p . vec r = 0`]

Options

A. Option1 ` hati +3 hatj - 2hatk`
B. Option2 `3hati + hatj + 2hatk`
C. Option3 `-3hati - hatj - 2hatk`
D. Option4 `- hati +3 hatj + 2hatk`

Explanation

To find the vector parallel to the electric field at position \(\vec{r}\), given that \(\vec{p} \cdot \vec{r} = 0\), we can follow these steps:

### Step 1: Write down the given vectors
The vectors are given as:
\[ \vec{p} = -\hat{i} - 3\hat{j} + 2\hat{k} \]
\[ \vec{r} = \hat{i} + 3\hat{j} + 5\hat{k} \]

### Step 2: Verify the dot product condition
We need to check that \(\vec{p} \cdot \vec{r} = 0\):
\[ \vec{p} \cdot \vec{r} = (-1)(1) + (-3)(3) + (2)(5) \]
Calculating this:
\[ = -1 - 9 + 10 = 0 \]
Thus, the condition \(\vec{p} \cdot \vec{r} = 0\) is satisfied.

### Step 3: Understand the relationship between \(\vec{p}\) and the electric field \(\vec{E}\)
Since \(\vec{p} \cdot \vec{r} = 0\), it indicates that the point is on the equator of the dipole. The electric field \(\vec{E}\) at any point on the equator of a dipole is anti-parallel to the dipole moment \(\vec{p}\).

### Step 4: Find the vector parallel to the electric field
The vector that is parallel to the electric field \(\vec{E}\) can be expressed as:
\[ \vec{A} = -\vec{p} \]
Substituting the value of \(\vec{p}\):
\[ \vec{A} = -(-\hat{i} - 3\hat{j} + 2\hat{k}) = \hat{i} + 3\hat{j} - 2\hat{k} \]

### Step 5: Conclusion
Thus, the vector parallel to the electric field at position \(\vec{r}\) is:
\[ \vec{A} = \hat{i} + 3\hat{j} - 2\hat{k} \]