If `x(hat(i) + hat(j) + hat(k))` is a unit vector then x equals ?

Options

A. `(1)/(sqrt(3))`
B. `pm sqrt(3)`
C. `sqrt(3)`
D. `pm (1)/(sqrt(3))`

Answer

To solve the problem, we need to find the value of \( x \) such that the vector \( x(\hat{i} + \hat{j} + \hat{k}) \) is a unit vector.

### Step-by-Step Solution:

1. **Define the Vector**:
Let \( \mathbf{a} = x(\hat{i} + \hat{j} + \hat{k}) \). This can be rewritten as:
\[ \mathbf{a} = x\hat{i} + x\hat{j} + x\hat{k} \]

2. **Magnitude of the Vector**:
The magnitude of vector \( \mathbf{a} \) is given by:
\[ |\mathbf{a}| = \sqrt{(x)^2 + (x)^2 + (x)^2} \]
This simplifies to:
\[ |\mathbf{a}| = \sqrt{3x^2} \]

3. **Set the Magnitude to 1**:
Since \( \mathbf{a} \) is a unit vector, we have:
\[ |\mathbf{a}| = 1 \]
Therefore, we set up the equation:
\[ \sqrt{3x^2} = 1 \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ 3x^2 = 1 \]

5. **Solve for \( x^2 \)**:
Dividing both sides by 3 gives:
\[ x^2 = \frac{1}{3} \]

6. **Find \( x \)**:
Taking the square root of both sides, we find:
\[ x = \pm \sqrt{\frac{1}{3}} = \pm \frac{1}{\sqrt{3}} \]

### Final Answer:
Thus, the value of \( x \) is:
\[ x = \pm \frac{1}{\sqrt{3}} \]