If the points `(1,1,p) a n d (3, 0, 1)`be equidistant from the plane ` vecr cdot(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 given plane defined by the equation:

\[ \vec{r} \cdot (3 \hat{i} + 4 \hat{j} - 12 \hat{k}) + 13 = 0 \]

### Step 1: Identify the coefficients of the plane equation
The equation of the plane can be rewritten in the standard form \( ax + by + cz + d = 0 \), where:
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