Find the angle between the planes whose vector equations are ` vec r.(2 hat i+2 hat j-3 hat k)=5`and ` vec r.(3 hat i-3 hat j+5 hat k)=3`.

Explanation

To find the angle between the planes given by the vector equations \(\vec{r} \cdot (2\hat{i} + 2\hat{j} - 3\hat{k}) = 5\) and \(\vec{r} \cdot (3\hat{i} - 3\hat{j} + 5\hat{k}) = 3\), we can follow these steps:

### Step 1: Identify the normals of the planes
The normal vector of the first plane can be extracted from the first equation:
\[ \vec{n_1} = 2\hat{i} + 2\hat{j} - 3\hat{k} \]
The normal vector of the second plane can be extracted from the second equation:
...