Q.Reduce the equation $\vec{r} \cdot (3\hat{i} + 4\hat{j} + 12\hat{k}) = 78$ to normal form and hence find

[!TLDR] Divide the vector equation by $|\vec n|$ to reach the normal form $\vec r\cdot\hat n=p$; $p$ is the perpendicular distance and the coefficients of $\hat n$ are the direction cosines. …

The given equation is $\vec r\cdot(3\hat i+4\hat j+12\hat k)=78$, so $\vec n = 3\hat i+4\hat j+12\hat k$ and

Dividing the equation through by $13$ gives the normal form $\vec r\cdot\hat n = p$:

(i) By Theorem 6.12, the right-hand side $p=6$ is the length of the perpendicular from the origin to the plane. …

1. Identify $\vec n$ from the vector equation $\vec r\cdot\vec n = d$.
2. Compute $|\vec n|$.
3. Divide the equation through by $|\vec n|$ so the right side becomes $d/|\vec n|$ and the left side involves the unit vector $\hat n$. …

Leaving the answer as $78/13$ instead of simplifying to $6$. Also forgetting to divide the vector part by $13$ as well (only simplifying the right-hand side), which leaves the norma …