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 vector components
The given vector can be expressed in terms of its components:
\[ \vec{A} = \hat{i} + \hat{j} + \sqrt{2} \hat{k} \]
From this, we can identify the components:
- \( A_x = 1 \) (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 of the vector \( \vec{A} \) 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 \( l, m, n \) are given by:
\[ l = \frac{A_x}{|\vec{A}|}, \quad m = \frac{A_y}{|\vec{A}|}, \quad n = \frac{A_z}{|\vec{A}|} \]
Substituting the values we found:
\[ l = \frac{1}{2}, \quad m = \frac{1}{2}, \quad n = \frac{\sqrt{2}}{2} \]

### Step 4: Write the final result
Thus, the direction cosines of the vector \( \hat{i} + \hat{j} + \sqrt{2} \hat{k} \) are:
\[ \left( \frac{1}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2} \right) \]