Find the direction ratios of orthogonal projection of line `(x-1)/1=(y+1)/(-2)=(z-2)/3` in the plane `x-y+2y-3=0.` also find the direction ratios of the image of the line in the plane.

Explanation

To solve the problem of finding the direction ratios of the orthogonal projection of the line given by \(\frac{x-1}{1} = \frac{y+1}{-2} = \frac{z-2}{3}\) onto the plane defined by the equation \(x - y + 2z - 3 = 0\), and also to find the direction ratios of the image of the line in the plane, we will follow these steps:

### Step 1: Identify the Direction Ratios of the Given Line
The line is given in symmetric form:
\[ \frac{x-1}{1} = \frac{y+1}{-2} = \frac{z-2}{3} \]
From this, we can identify the direction ratios of the line as:
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