If a line makes an angle of $$\frac{\pi}{4}$$ with the positive directions of each of $$x$$-axis and $$y$$-axis, then the angle that the line makes with the positive direction of the $$z$$-axis is

1. Identify the Given Direction Angles

Let the angles made by the straight line with the positive directions of the $$x$$, $$y$$, and $$z$$ axes be $$\alpha$$, $$\beta$$, and $$\gamma$$ respectively.

We are given the following values from the problem:

$$\alpha = \frac{\pi}{4}$$

$$\beta = \frac{\pi}{4}$$

2. Recall the Direction Cosines Identity

For any straight line in a three-dimensional space, its direction cosines $$(l, m, n)$$ are defined as $$l = \cos \alpha$$, $$m = \cos \beta$$, and $$n = \cos \gamma$$. They satisfy the fundamental geometric identity:

$$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$$

3. Substitute the Known Values

Substitute the given angles into the direction cosine identity:

$$\cos^2\left(\frac{\pi}{4}\right) + \cos^2\left(\frac{\pi}{4}\right) + \cos^2 \gamma = 1$$

Since $$\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$$, their squares evaluate to:

$$\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + \cos^2 \gamma = 1$$

$$\frac{1}{2} + \frac{1}{2} + \cos^2 \gamma = 1$$

4. Solve for the Unknown Angle

Combine the numeric fractions on the left-hand side:

$$1 + \cos^2 \gamma = 1$$

$$\cos^2 \gamma = 1 - 1$$

$$\cos^2 \gamma = 0$$

Taking the square root of both sides gives:

$$\cos \gamma = 0$$

Find the principal angle value within the standard range $$[0, \pi]$$ where the cosine vanishes:

$$\gamma = \frac{\pi}{2}$$

Final Answer

The angle that the line makes with the positive direction of the z-axis is $$\frac{\pi}{2}$$.