Find the direction cosines of the resultant of the vectors `(hati+hatj+hatk),(-hati+hatj+hatk),(hati-hatj+hatk) and (hati+hatj-hatk)`.

Answer

To find the direction cosines of the resultant of the vectors \( \hat{i} + \hat{j} + \hat{k} \), \( -\hat{i} + \hat{j} + \hat{k} \), \( \hat{i} - \hat{j} + \hat{k} \), and \( \hat{i} + \hat{j} - \hat{k} \), we will follow these steps:

### Step 1: Define the Vectors
Let:
- \( \vec{A} = \hat{i} + \hat{j} + \hat{k} \)
- \( \vec{B} = -\hat{i} + \hat{j} + \hat{k} \)
- \( \vec{C} = \hat{i} - \hat{j} + \hat{k} \)
- \( \vec{D} = \hat{i} + \hat{j} - \hat{k} \)

### Step 2: Find the Resultant Vector
The resultant vector \( \vec{R} \) is given by the sum of the vectors:
\[ \vec{R} = \vec{A} + \vec{B} + \vec{C} + \vec{D} \]
Substituting the vectors:
\[ \vec{R} = (\hat{i} + \hat{j} + \hat{k}) + (-\hat{i} + \hat{j} + \hat{k}) + (\hat{i} - \hat{j} + \hat{k}) + (\hat{i} + \hat{j} - \hat{k}) \]

### Step 3: Simplify the Resultant Vector
Now, let's combine the components:
- The \( \hat{i} \) components: \( 1 - 1 + 1 + 1 = 2 \)
- The \( \hat{j} \) components: \( 1 + 1 - 1 + 1 = 2 \)
- The \( \hat{k} \) components: \( 1 + 1 + 1 - 1 = 2 \)

Thus, we have:
\[ \vec{R} = 2\hat{i} + 2\hat{j} + 2\hat{k} \]

### Step 4: Calculate the Magnitude of the Resultant Vector
The magnitude of \( \vec{R} \) is given by:
\[ |\vec{R}| = \sqrt{(2)^2 + (2)^2 + (2)^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3} \]

### Step 5: Find the Unit Vector in the Direction of \( \vec{R} \)
The unit vector \( \hat{r} \) in the direction of \( \vec{R} \) is given by:
\[ \hat{r} = \frac{\vec{R}}{|\vec{R}|} = \frac{2\hat{i} + 2\hat{j} + 2\hat{k}}{2\sqrt{3}} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} \]

### Step 6: Determine the Direction Cosines
The direction cosines \( l, m, n \) are the components of the unit vector \( \hat{r} \):
\[ l = \frac{1}{\sqrt{3}}, \quad m = \frac{1}{\sqrt{3}}, \quad n = \frac{1}{\sqrt{3}} \]

### Conclusion
The direction cosines of the resultant vector are:
\[ \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \]

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