If the points `(1,1,p) a n d (3, 0, 1)`be equidistant from the plane ` vecr *(3 hat i+4 hat j-12 hat k)+13=0`, then find the value of p.

Explanation

To solve the problem, we need to find the value of \( p \) such that the points \( (1, 1, p) \) and \( (3, 0, 1) \) are equidistant from the plane defined by the equation \( \vec{r} \cdot (3\hat{i} + 4\hat{j} - 12\hat{k}) + 13 = 0 \).

### Step 1: Write the equation of the plane in Cartesian form
The equation of the plane can be rewritten as:
\[ 3x + 4y - 12z + 13 = 0 \]

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