If `hat(a)` is a unit vector and projection of x along `hat(a)` is 2 units and `(hat(a)times x)+b=x,` then x is equal to

Options

A. `(1)/(2)(hat(a)-b+(hatatimesb))`
B. `(1)/(2)(2hat(a)+b+(hatatimesb))`
C. `(hat(a)+(hatatimesb))`
D. None of these

Answer

To solve the problem step by step, we will analyze the given information and use vector operations to find the vector \( x \).

### Step 1: Understand the given information
We know that:
- \( \hat{a} \) is a unit vector.
- The projection of \( x \) along \( \hat{a} \) is 2 units.
- The equation \( \hat{a} \times x + b = x \) holds.

### Step 2: Write the projection of \( x \) along \( \hat{a} \)
The projection of vector \( x \) along the unit vector \( \hat{a} \) is given by:
\[ \text{Projection of } x \text{ along } \hat{a} = x \cdot \hat{a} \]
Since the projection is given as 2 units, we have:
\[ x \cdot \hat{a} = 2 \]

### Step 3: Rearrange the given equation
The equation \( \hat{a} \times x + b = x \) can be rearranged to isolate the cross product:
\[ \hat{a} \times x = x - b \]

### Step 4: Take the cross product with \( \hat{a} \)
Taking the cross product of both sides with \( \hat{a} \):
\[ \hat{a} \times (\hat{a} \times x) = \hat{a} \times (x - b) \]

### Step 5: Use the vector triple product identity
Using the vector triple product identity \( \hat{a} \times (\hat{a} \times x) = (\hat{a} \cdot x) \hat{a} - (\hat{a} \cdot \hat{a}) x \):
\[ (\hat{a} \cdot x) \hat{a} - x = \hat{a} \times (x - b) \]
Since \( \hat{a} \cdot \hat{a} = 1 \), we can substitute:
\[ 2 \hat{a} - x = \hat{a} \times (x - b) \]

### Step 6: Substitute the value of \( x \)
Now, we can express \( x \) in terms of \( \hat{a} \) and \( b \):
\[ x = 2 \hat{a} + \hat{a} \times (x - b) \]

### Step 7: Solve for \( x \)
We can rearrange the equation:
\[ x - \hat{a} \times (x - b) = 2 \hat{a} \]
Let’s denote \( x - b = y \), then:
\[ x = 2 \hat{a} + \hat{a} \times y \]
Substituting back gives us:
\[ x = 2 \hat{a} + b + \hat{a} \times (x - b) \]

### Step 8: Final expression for \( x \)
After simplification, we find:
\[ x = 2 \hat{a} + b + \hat{a} \times b \]

### Conclusion
Thus, the expression for \( x \) is:
\[ \boxed{x = 2 \hat{a} + b + \hat{a} \times b} \]