Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes ` vecr cdot( hat i- hat j+2 hat k)=5`and ` vecr cdot(3 hat i+ hat j+ hat k)=6`.

Explanation

To find the vector equation of the line passing through the point \( (1, 2, 3) \) and parallel to the given planes, we can follow these steps:

### Step 1: Identify the normal vectors of the planes
The equations of the planes are given as:
1. \( \vec{r} \cdot (\hat{i} - \hat{j} + 2\hat{k}) = 5 \)
2. \( \vec{r} \cdot (3\hat{i} + \hat{j} + \hat{k}) = 6 \)

From these equations, we can identify the normal vectors of the planes:
...