If `a=2hati+2hatj-hatk and |xveca|=1`, then find x.

Answer

To solve the problem, we need to find the value of \( x \) given that \( \vec{a} = 2\hat{i} + 2\hat{j} - \hat{k} \) and \( |\vec{x} \vec{a}| = 1 \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We are given a vector \( \vec{a} = 2\hat{i} + 2\hat{j} - \hat{k} \) and the condition \( |\vec{x} \vec{a}| = 1 \).

2. **Using the Magnitude Property**:
The expression \( |\vec{x} \vec{a}| \) can be rewritten using the property of magnitudes:
\[ |\vec{x} \vec{a}| = |\vec{x}| |\vec{a}| \]
Therefore, we have:
\[ |\vec{x}| |\vec{a}| = 1 \]

3. **Finding the Magnitude of Vector \( \vec{a} \)**:
To find \( |\vec{a}| \), we calculate:
\[ |\vec{a}| = \sqrt{(2)^2 + (2)^2 + (-1)^2} \]
Simplifying this:
\[ |\vec{a}| = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

4. **Substituting Back into the Magnitude Equation**:
Now substituting \( |\vec{a}| \) back into the equation:
\[ |\vec{x}| \cdot 3 = 1 \]

5. **Solving for \( |\vec{x}| \)**:
Dividing both sides by 3 gives:
\[ |\vec{x}| = \frac{1}{3} \]

6. **Conclusion**:
Thus, the value of \( x \) is:
\[ x = \frac{1}{3} \]

### Final Answer:
\[ x = \frac{1}{3} \]