Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane ` vec r.( hat i+2 hat j-5 hat k)+9=0`.

Explanation

To find the vector equation of the line passing through the point (1, 2, 3) and perpendicular to the plane given by the equation \(\vec{r} \cdot (\hat{i} + 2\hat{j} - 5\hat{k}) + 9 = 0\), we can follow these steps:

### Step 1: Identify the normal vector of the plane
The equation of the plane can be rewritten in the form \(\vec{r} \cdot \vec{n} + d = 0\), where \(\vec{n}\) is the normal vector of the plane. From the equation \(\vec{r} \cdot (\hat{i} + 2\hat{j} - 5\hat{k}) + 9 = 0\), we can identify the normal vector:
\[ \vec{n} = \hat{i} + 2\hat{j} - 5\hat{k} \]

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