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Step-by-Step Solution
Step 1: Understand the problem
A line in three-dimensional space makes an angle of $ \frac{\pi}{4} $ with the positive x-axis and also $ \frac{\pi}{4} $ with the positive y-axis. We want to find the angle $ \alpha $ that this line makes with the positive z-axis.
Step 2: Recall the property of direction cosines
If a line makes angles $ \alpha_x $, $ \alpha_y $, and $ \alpha_z $ with the positive x-axis, y-axis, and z-axis respectively, its direction cosines are:
$ l = \cos \alpha_x,\quad m = \cos \alpha_y,\quad n = \cos \alpha_z. $
These direction cosines satisfy the relation:
$ l^2 + m^2 + n^2 = 1.
Step 3: Substitute known angles and set up the equation
Here, $ \alpha_x = \frac{\pi}{4} $ and $ \alpha_y = \frac{\pi}{4} $. Let $ \alpha_z = \alpha $. Therefore,
$ l = \cos \frac{\pi}{4},\quad m = \cos \frac{\pi}{4},\quad n = \cos \alpha.
Substitute these into the direction cosines relation:
$ \cos^2 \frac{\pi}{4} + \cos^2 \frac{\pi}{4} + \cos^2 \alpha = 1.
Step 4: Evaluate and solve for $ \alpha $
$ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \text{so } \cos^2 \frac{\pi}{4} = \frac{1}{2}.
Hence, the equation becomes:
$ \frac{1}{2} + \frac{1}{2} + \cos^2 \alpha = 1.
$ 1 + \cos^2 \alpha = 1 \quad \Longrightarrow \quad \cos^2 \alpha = 0.
Thus, $ \cos \alpha = 0 $, which gives:
$ \alpha = \frac{\pi}{2},
(since we are looking for an angle between 0 and $ \pi $ with the positive direction of an axis).
Step 5: Conclude the answer
The angle that the line makes with the positive direction of the z-axis is therefore:
$ \boxed{\frac{\pi}{2}}.
