If three vector along coordinate axis represent the adjacent sides of a cube of length b, then the unit vector along its diaonal passing thourth the origin will be

Options

A. `(hati+hatj+hatk)/(sqrt(2))`
B. `(hati+hatj+hatk)/(sqrt(36))`
C. `hati+hatj+hatk`
D. `(hati+hatj+hatk)/(sqrt(3))`

Explanation

To find the unit vector along the diagonal of a cube with side length \( b \), we can follow these steps:

### Step 1: Understand the Vectors Representing the Cube
The cube has three adjacent sides along the coordinate axes. We can represent these sides using vectors:
- Along the x-axis: \( \vec{A} = b \hat{i} \)
- Along the y-axis: \( \vec{B} = b \hat{j} \)
- Along the z-axis: \( \vec{C} = b \hat{k} \)

### Step 2: Find the Diagonal Vector
The diagonal vector \( \vec{D} \) of the cube can be obtained by adding the vectors representing the sides:
\[ \vec{D} = \vec{A} + \vec{B} + \vec{C} = b \hat{i} + b \hat{j} + b \hat{k} \]
This simplifies to:
\[ \vec{D} = b (\hat{i} + \hat{j} + \hat{k}) \]

### Step 3: Calculate the Magnitude of the Diagonal Vector
The magnitude of the diagonal vector \( \vec{D} \) is calculated as follows:
\[ |\vec{D}| = \sqrt{(b)^2 + (b)^2 + (b)^2} = \sqrt{3b^2} = b\sqrt{3} \]

### Step 4: Find the Unit Vector Along the Diagonal
The unit vector \( \hat{d} \) in the direction of the diagonal is given by dividing the diagonal vector by its magnitude:
\[ \hat{d} = \frac{\vec{D}}{|\vec{D}|} = \frac{b (\hat{i} + \hat{j} + \hat{k})}{b\sqrt{3}} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} \]

### Final Answer
Thus, the unit vector along the diagonal passing through the origin is:
\[ \hat{d} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} \]