Find the values of 'x' for which `x(hat(i)+hat(j)+hat(k))` is a unit vector.

Answer

To find the values of 'x' for which the vector \( \vec{A} = x(\hat{i} + \hat{j} + \hat{k}) \) is a unit vector, we can follow these steps:

### Step 1: Define the vector
We start by defining the vector:
\[ \vec{A} = x(\hat{i} + \hat{j} + \hat{k}) = x\hat{i} + x\hat{j} + x\hat{k} \]

### Step 2: Find the magnitude of the vector
The magnitude of the vector \( \vec{A} \) is given by:
\[ |\vec{A}| = \sqrt{(x)^2 + (x)^2 + (x)^2} = \sqrt{3x^2} \]

### Step 3: Set the magnitude equal to 1
Since we want \( \vec{A} \) to be a unit vector, we set the magnitude equal to 1:
\[ \sqrt{3x^2} = 1 \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:
\[ 3x^2 = 1 \]

### Step 5: Solve for \( x^2 \)
Now, we can solve for \( x^2 \):
\[ x^2 = \frac{1}{3} \]

### Step 6: Find the values of \( x \)
Taking the square root of both sides gives us:
\[ x = \pm \frac{1}{\sqrt{3}} \]

### Final Result
Thus, the values of \( x \) for which \( \vec{A} \) is a unit vector are:
\[ x = \frac{1}{\sqrt{3}} \quad \text{and} \quad x = -\frac{1}{\sqrt{3}} \]

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