Find the vector equation of the line passing through the point (1,2,3) and parallel to the planes ` vec rdot( hat i- hat j+2 hat k)=5` and ` vec 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 point and the normal vectors of the planes
The line passes through the point \( P(1, 2, 3) \). 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:
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