Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes ` -> rdot( hat i- hat j+2 hat k)=5`and ` -> rdot(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 in the form \( \mathbf{r} \cdot \mathbf{n} = d \), where \( \mathbf{n} \) is the normal vector of the plane.

1. For the first plane \( \mathbf{r} \cdot (\hat{i} - \hat{j} + 2\hat{k}) = 5 \), the normal vector \( \mathbf{n_1} = (1, -1, 2) \).
2. For the second plane \( \mathbf{r} \cdot (3\hat{i} + \hat{j} + \hat{k}) = 6 \), the normal vector \( \mathbf{n_2} = (3, 1, 1) \).

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