If a line makes angles `(pi)/(2), (3pi)/(4) and (pi)/(4)` with x,y,z axis respectively, then direction cosines of this line are :

Options

A. `pm` (1,1,1)
B. `pm (0, (1)/(sqrt(2)), (1)/(sqrt(2)))`
C. `pm ((1)/(2), (1)/(2), (1)/(2))`
D. `pm (0, -(1)/(sqrt(2)), (1)/(sqrt(2)))`

Answer

To find the direction cosines of a line that makes angles \(\frac{\pi}{2}\), \(\frac{3\pi}{4}\), and \(\frac{\pi}{4}\) with the x, y, and z axes respectively, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Angles:**
- Let \(\alpha\) be the angle with the x-axis.
- Let \(\beta\) be the angle with the y-axis.
- Let \(\gamma\) be the angle with the z-axis.

From the question, we have:
\[ \alpha = \frac{\pi}{2}, \quad \beta = \frac{3\pi}{4}, \quad \gamma = \frac{\pi}{4} \]

2. **Calculate Direction Cosines:**
The direction cosines \(l\), \(m\), and \(n\) are given by:
\[ l = \cos(\alpha), \quad m = \cos(\beta), \quad n = \cos(\gamma) \]

Now, we will calculate each direction cosine:

- **For \(l\):**
\[ l = \cos\left(\frac{\pi}{2}\right) = 0 \]

- **For \(m\):**
\[ m = \cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}} \quad \text{(since } \cos\left(\frac{3\pi}{4}\right) = -\cos\left(\frac{\pi}{4}\right)\text{)} \]

- **For \(n\):**
\[ n = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

3. **Write the Direction Cosines:**
Now we can summarize the direction cosines:
\[ (l, m, n) = \left(0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \]

### Final Answer:
The direction cosines of the line are:
\[ \boxed{\left(0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)} \]