1. Key Concepts and Formulas

• Direction Cosines: For a line in three-dimensional space, the angles it makes with the positive directions of the $x$-axis, $y$-axis, and $z$-axis are conventionally denoted by $\alpha$, $\beta$, and $\gamma$, respectively. The cosines of these angles, $\cos \alpha$, $\cos \beta$, and $\cos \gamma$, are called the direction cosines of the line. They are often represented by $l$, $m$, and $n$.
  • $l = \cos \alpha$
  • $m = \cos \beta$
  • $n = \cos \gamma$
• Fundamental Identity of Direction Cosines: A crucial property of direction cosines is that the sum of their squares is always equal to 1. This identity arises from the fact that $(l, m, n)$ represents a unit vector along the line's direction. $l^2 + m^2 + n^2 = 1$ or equivalently, $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$

2. Step-by-Step Solution

Step 1: Identify Given Information and the Unknown The problem provides information about the angles a line makes with the positive $x$-axis and $y$-axis.

• The angle with the positive $x$-axis is given as $\alpha = \pi/4$.
• The angle with the positive $y$-axis is given as $\beta = \pi/4$.
• We need to determine the angle the line makes with the positive $z$-axis, which we denote as $\gamma$.

Step 2: Express Direction Cosines in Terms of Given Angles Using the definition of direction cosines from Key Concepts:

• The direction cosine with respect to the $x$-axis is $l = \cos \alpha = \cos(\pi/4)$.
• The direction cosine with respect to the $y$-axis is $m = \cos \beta = \cos(\pi/4)$.
• The direction cosine with respect to the $z$-axis is $n = \cos \gamma$.

Step 3: Apply the Fundamental Identity of Direction Cosines The fundamental identity states that the sum of the squares of the direction cosines is 1. We substitute the expressions for $l$, $m$, and $n$ from Step 2 into this identity: $l^2 + m^2 + n^2 = 1$ $\cos^2(\pi/4) + \cos^2(\pi/4) + \cos^2 \gamma = 1$

Step 4: Substitute Known Trigonometric Values and Solve for the Unknown Angle First, recall the exact value of $\cos(\pi/4)$: $\cos(\pi/4) = \frac{1}{\sqrt{2}}$ Now, substitute this value into the equation from Step 3: $\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + \cos^2 \gamma = 1$ Calculate the squares of the known terms: $\frac{1}{2} + \frac{1}{2} + \cos^2 \gamma = 1$ Simplify the sum of the numerical terms on the left side: $1 + \cos^2 \gamma = 1$ To find $\cos^2 \gamma$, subtract 1 from both sides of the equation: $\cos^2 \gamma = 1 - 1$ $\cos^2 \gamma = 0$ Take the square root of both sides to find $\cos \gamma$: $\cos \gamma = 0$ Finally, we need to find the angle $\gamma$ whose cosine is 0. By convention, the angles a line makes with the coordinate axes are taken in the range $[0, \pi]$ radians (or $0^\circ$ to $180^\circ$). In this range, the unique angle whose cosine is 0 is $\pi/2$. $\gamma = \frac{\pi}{2}$

3. Common Mistakes & Tips

• Forgetting the Identity: The identity $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ is the cornerstone of many 3D geometry problems involving lines. Make sure to commit it to memory.
• Trigonometric Values: Accurately recalling standard trigonometric values (like $\cos(\pi/4) = 1/\sqrt{2}$) is essential for efficient problem-solving.
• Range of Angles: Remember that the angles $\alpha, \beta, \gamma$ are typically considered in the range $[0, \pi]$. This helps in uniquely determining the angle when solving for its cosine (e.g., if $\cos \theta = 0$, then $\theta = \pi/2$ is the unique solution in this range).
• Physical Interpretation: An angle of $\pi/2$ with the $z$-axis means the line is perpendicular to the $z$-axis. Since it makes equal angles with the $x$ and $y$ axes, this implies the line lies in a plane parallel to the $xy$-plane and makes equal angles with $x$ and $y$ axes in that plane.

4. Summary

This problem is a direct application of the fundamental identity involving the direction cosines of a line in three-dimensional space: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. By substituting the given angles for the $x$-axis and $y$-axis, we calculated their respective squared direction cosines. The sum of these values, along with the unknown squared direction cosine for the $z$-axis, must equal 1. Solving the resulting equation yielded $\cos \gamma = 0$, which implies $\gamma = \pi/2$.

5. Final Answer

The angle that the line makes with the positive direction of the $z$-axis is $\frac{\pi}{2}$, which corresponds to option (B).

The final answer is $\boxed{\text{\pi / 2}}$.