Equation of line BA is `x+y=1`. Find a unit vector along the reflected ray AC.

A. `((hat(i)+hat(j)))/sqrt(2)`
B. `((hat(i)-hat(j)))/sqrt(2)`
C. `sqrt(2)(hat(i)+hat(j))`
D. None

Options

A. `((hat(i)+hat(j)))/sqrt(2)`
B. `((hat(i)-hat(j)))/sqrt(2)`
C. `sqrt(2)(hat(i)+hat(j))`
D. None

Explanation

To find the unit vector along the reflected ray AC given the equation of line BA as \( x + y = 1 \), we can follow these steps:

### Step 1: Understand the given line
The equation \( x + y = 1 \) represents a line in the Cartesian plane. We can rewrite this in slope-intercept form:
\[ y = -x + 1 \]
From this, we can see that the slope of line BA is -1.

### Step 2: Determine the angle of incidence
The angle of incidence is the angle between the incoming ray (line BA) and the normal to the line. Since the slope of line BA is -1, the angle it makes with the x-axis can be calculated using the tangent function:
\[ \tan(\theta) = -1 \implies \theta = 135^\circ \text{ (since it lies in the second quadrant)} \]

### Step 3: Find the angle of reflection
According to the law of reflection, the angle of reflection is equal to the angle of incidence. Therefore, the angle AC makes with the normal will also be \( 135^\circ \).

### Step 4: Calculate the direction of AC
The direction of the reflected ray AC can be determined by considering that it will make an angle of \( 45^\circ \) with the x-axis (since it reflects off the line BA). Thus, the direction of AC can be represented as:
\[ \text{Unit vector along AC} = \cos(45^\circ) \hat{i} + \sin(45^\circ) \hat{j} \]

### Step 5: Substitute the values of cosine and sine
We know that:
\[ \cos(45^\circ) = \sin(45^\circ) = \frac{1}{\sqrt{2}} \]
Substituting these values gives:
\[ \text{Unit vector along AC} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]

### Step 6: Identify the correct option
Now we can compare this result with the given options. The unit vector along AC is:
\[ \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]
This matches with option A.

### Final Answer:
The unit vector along the reflected ray AC is:
\[ \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]