Find the value of `mu` if the points with position vectors ` hat i - hat j + 3 hat k` and ` 3 hat i + 4 hat j+ mu hat k` are equidistant form the plane ` hat r .(5 hat i + 2 hat j - 7 hat k )+ 8 =0`

Explanation

To find the value of `mu` such that the points with position vectors \(\hat{i} - \hat{j} + 3\hat{k}\) and \(3\hat{i} + 4\hat{j} + \mu\hat{k}\) are equidistant from the plane given by the equation \(\hat{r} \cdot (5\hat{i} + 2\hat{j} - 7\hat{k}) + 8 = 0\), we can follow these steps:

### Step 1: Identify the normal vector and the distance formula
The normal vector \(\hat{n}\) of the plane can be identified from the equation of the plane:
\[ \hat{n} = 5\hat{i} + 2\hat{j} - 7\hat{k} \]
The distance \(d\) from a point with position vector \(\hat{r_0}\) to the plane is given by:
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