The direction cosines of a vector `hati+hatj+sqrt(2) hatk` are:-

Explanation

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:
\[ \vec{A} = \hat{i} + \hat{j} + \sqrt{2} \hat{k} \]
Here, the components are:
- \( A_x = 1 \) (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 \( \vec{A} \) is calculated using the formula:
\[ |\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} \]
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 the formulas:
\[ l = \frac{A_x}{|\vec{A}|}, \quad m = \frac{A_y}{|\vec{A}|}, \quad n = \frac{A_z}{|\vec{A}|} \]
Now substituting the values:
\[ l = \frac{1}{2}, \quad m = \frac{1}{2}, \quad n = \frac{\sqrt{2}}{2} \]

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

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