Problem 58 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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 (A) \(\frac{\pi}{6}\) (B) \(\frac{\pi}{3}\) (C) \(\frac{\pi}{4}\) (D) \(\frac{\pi}{2}\)

Step-by-Step Solution

Answer

The angle the line makes with the positive direction of the z-axis is \( \frac{\pi}{2} \) (Option D).

Step 1: Understanding the Problem

We need to find the angle the line makes with the positive direction of the z-axis given that it makes an angle of \( \frac{\pi}{4} \) with both the x-axis and y-axis.

Step 2: Introduction to Direction Cosines

The direction cosines of a line are defined as \( l \), \( m \), and \( n \), corresponding to the angles \( \alpha \), \( \beta \), and \( \gamma \) it makes with the x, y, and z-axes, respectively. It follows that \( l = \cos \alpha \), \( m = \cos \beta \), and \( n = \cos \gamma \).

Step 3: Find Direction Cosines for X and Y Axes

Since the line makes an angle of \( \frac{\pi}{4} \) with both x and y axes, we have \( l = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) and \( m = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 4: Use the Direction Cosines Equation

The direction cosines satisfy the equation \( l^2 + m^2 + n^2 = 1 \). Substitute the known values: \( \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 + n^2 = 1 \).

Step 5: Calculate n

Simplify the equation: \( \frac{1}{2} + \frac{1}{2} + n^2 = 1 \) giving \( n^2 = 0 \).

Step 6: Find \( \gamma \)

Since \( n = 0 \), \( \cos \gamma = 0 \) implies \( \gamma = \frac{\pi}{2} \) or \( \gamma = \frac{3\pi}{2} \). Thus, \( \gamma = \frac{\pi}{2} \) is the angle with the positive z-axis.

Key Concepts

Angle Between Line and Axes

When dealing with lines in three-dimensional space, it is crucial to understand how they interact with the coordinate axes. Specifically, the angle a line makes with each axis—known as direction angles—can tell us a lot about its orientation.
Consider a line making an angle of \(\alpha = \frac{\pi}{4}\) with both the \(x\)-axis and the \(y\)-axis. These angles help determine the line's direction in the 3D space.

• The line's angle with the \(z\)-axis remains to be calculated using the relationship among the direction cosines.
• The direction cosine equation, \(l^2 + m^2 + n^2 = 1\), ties together the squares of the cosines of the angles with the axes.

This relationship ensures that the sum of the squares of these cosines equals one, enforcing the geometric constraint of 3D lines.

Direction Angles

Direction angles \((\alpha, \beta, \gamma)\) are the angles a line makes with the positive directions of the \(x\), \(y\), and \(z\) axes respectively. These angles are pivotal when using direction cosmics to uniquely define a line's direction in space.
In the given example, we know:

• \(\alpha = \frac{\pi}{4}\) for the \(x\)-axis, giving us the direction cosine \(l = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).
• Similarly, \(\beta = \frac{\pi}{4}\) for the \(y\)-axis results in the direction cosine \(m = \frac{\sqrt{2}}{2}\).

To find the angle with the \(z\)-axis \(\gamma\), the remaining direction cosine \(n\) is calculated from \(l^2 + m^2 + n^2 = 1\).
This exercise illustrates how if \(n = 0\), it implies \(\gamma = \frac{\pi}{2}\); thus, the line makes a right angle with the \(z\)-axis.

3D Geometry

Three-dimensional geometry is a part of mathematics that deals with the positioning and orientation of shapes in space. Understanding 3D geometry involves visualizing how lines interact with three axes: \(x\), \(y\), and \(z\).
This field considers several aspects:

• The spatial relationships involve direction cosines and direction angles, assisting in determining a line's orientation.
• 3D vectors and equations help define lines and planes, integral for understanding spatial problems like determining angles with axes.

In our example, knowing that the line makes a \(\frac{\pi}{4}\) angle with both \(x\) and \(y\) axes simplifies solving for its orientation relative to \(z\).
By systematically using the properties of direction cosines, we can visualize and solve how the line interacts with the axes—showing that its layout is neatly perpendicular to the \(z\)-axis at \(\gamma = \frac{\pi}{2}\).