A ray of light is incident on a plane mirror along a vector `hat i+hat j- hat k.`
The normal on incidence point is along `hat i+ hat j `.Find a unit vector along the
reflected ray.

Explanation

To find the unit vector along the reflected ray when a ray of light is incident on a plane mirror, we can follow these steps:

### Step 1: Identify the Incident Vector and Normal Vector
The incident ray is given by the vector:
\[ \vec{I} = \hat{i} + \hat{j} - \hat{k} \]
The normal vector at the point of incidence is given by:
\[ \vec{N} = \hat{i} + \hat{j} \]

### Step 2: Normalize the Normal Vector
To reflect the incident ray, we first need to normalize the normal vector. The magnitude of the normal vector \(\vec{N}\) is calculated as follows:
\[ |\vec{N}| = \sqrt{(1^2 + 1^2)} = \sqrt{2} \]
Thus, the unit normal vector \(\hat{n}\) is:
\[ \hat{n} = \frac{\vec{N}}{|\vec{N}|} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]

### Step 3: Use the Reflection Formula
The reflection of a vector \(\vec{I}\) about a normal vector \(\hat{n}\) is given by:
\[ \vec{R} = \vec{I} - 2(\vec{I} \cdot \hat{n})\hat{n} \]
First, we need to compute the dot product \(\vec{I} \cdot \hat{n}\):
\[ \vec{I} \cdot \hat{n} = (\hat{i} + \hat{j} - \hat{k}) \cdot \left(\frac{\hat{i} + \hat{j}}{\sqrt{2}}\right) = \frac{1 + 1 + 0}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Step 4: Calculate the Reflected Vector
Now we can substitute this back into the reflection formula:
\[ \vec{R} = \vec{I} - 2(\sqrt{2})\hat{n} \]
Substituting \(\hat{n}\):
\[ \vec{R} = \hat{i} + \hat{j} - \hat{k} - 2\sqrt{2}\left(\frac{\hat{i} + \hat{j}}{\sqrt{2}}\right) \]
This simplifies to:
\[ \vec{R} = \hat{i} + \hat{j} - \hat{k} - 2(\hat{i} + \hat{j}) = \hat{i} + \hat{j} - \hat{k} - 2\hat{i} - 2\hat{j} \]
\[ \vec{R} = -\hat{i} - \hat{j} - \hat{k} \]

### Step 5: Find the Unit Vector Along the Reflected Ray
Now we need to find the unit vector along the reflected ray \(\vec{R}\):
\[ |\vec{R}| = \sqrt{(-1)^2 + (-1)^2 + (-1)^2} = \sqrt{3} \]
Thus, the unit vector \(\hat{R}\) along the reflected ray is:
\[ \hat{R} = \frac{\vec{R}}{|\vec{R}|} = \frac{-\hat{i} - \hat{j} - \hat{k}}{\sqrt{3}} \]

### Final Answer
The unit vector along the reflected ray is:
\[ \hat{R} = \frac{-\hat{i} - \hat{j} - \hat{k}}{\sqrt{3}} \]