The direction cosines of a vector `hat(i) + hat(j) + sqrt(2)hat(k)` are :

Options

A. `(1)/(2),(1)/(2),1`
B. `(1)/(sqrt(2)),(1)/(sqrt(2)),(1)/(2)`
C. `(1)/(2),(1)/(2),(1)/(sqrt(2))`
D. `(1)/(sqrt(2)),(1)/(sqrt(2)),(1)/(sqrt(2))`

Answer

To find the direction cosines of the vector \(\hat{i} + \hat{j} + \sqrt{2}\hat{k}\), we will follow these steps:

### Step 1: Identify the components of the vector
The vector can be expressed in component form as:
\[ \vec{A} = (1, 1, \sqrt{2}) \]
where:
- \(A_x = 1\) (coefficient of \(\hat{i}\))
- \(A_y = 1\) (coefficient of \(\hat{j}\))
- \(A_z = \sqrt{2}\) (coefficient of \(\hat{k}\))

### Step 2: Calculate the magnitude of the vector
The magnitude \( |\vec{A}| \) of the vector is given by:
\[ |\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} \]
Substituting the values:
\[ |\vec{A}| = \sqrt{1^2 + 1^2 + (\sqrt{2})^2} = \sqrt{1 + 1 + 2} = \sqrt{4} = 2 \]

### Step 3: Calculate the direction cosines
The direction cosines \( \cos \alpha, \cos \beta, \cos \gamma \) are defined as:
\[ \cos \alpha = \frac{A_x}{|\vec{A}|}, \quad \cos \beta = \frac{A_y}{|\vec{A}|}, \quad \cos \gamma = \frac{A_z}{|\vec{A}|} \]

Substituting the values we found:
- For \( \cos \alpha \):
\[ \cos \alpha = \frac{1}{2} \]
- For \( \cos \beta \):
\[ \cos \beta = \frac{1}{2} \]
- For \( \cos \gamma \):
\[ \cos \gamma = \frac{\sqrt{2}}{2} \]

### Step 4: Write the direction cosines
Thus, the direction cosines of the vector \(\hat{i} + \hat{j} + \sqrt{2}\hat{k}\) are:
\[ \cos \alpha = \frac{1}{2}, \quad \cos \beta = \frac{1}{2}, \quad \cos \gamma = \frac{\sqrt{2}}{2} \]

### Final Answer
The direction cosines are:
\[ \left( \frac{1}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2} \right) \]

---