Problem 14 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A line \(\mathrm{AB}\) in three-dimensional space makes angles \(45^{\circ}\) and \(120^{\circ}\) with the positive \(x\)-axis and the positive \(y\)-axis respectively. If AB makes an acute angle \(\theta\) with the positive z-axis, then \(\theta\) equals [2010] (a) \(45^{\circ}\) (b) \(60^{\circ}\) (c) \(75^{\circ}\) (d) \(30^{\circ}\)

Step-by-Step Solution

Answer

The acute angle \(\theta\) with the z-axis is \(60^{\circ}\).

Step 1: Understanding Angles in 3D Space

In 3D geometry, a line can be represented by its direction cosines, which are the cosines of the angles that the line makes with the positive x, y, and z axes. Based on the problem, these angles are given for x and y axes as \(45^{\circ}\) and \(120^{\circ}\) respectively.

Step 2: Calculating Direction Cosines

The direction cosine for the x-axis, \( \cos(\alpha) \), is \( \cos(45^{\circ}) = \frac{1}{\sqrt{2}} \). For the y-axis, \( \cos(\beta) \), is \( \cos(120^{\circ}) = -\frac{1}{2} \). We need to find \( \cos(\theta) \) for the z-axis.

Step 3: Applying the Direction Cosines Property

The sum of the squares of the direction cosines is 1: \( \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\theta) = 1 \). Thus, substitute the known values: \( \left(\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{2}\right)^2 + \cos^2(\theta) = 1 \).

Step 4: Solving the Equation

Calculate each part: \( \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \). Thus, \( \cos^2(\theta) = 1 - \frac{3}{4} = \frac{1}{4} \). Solve for \(\cos(\theta)\): \(\cos(\theta) = \pm \frac{1}{2}\).

Step 5: Determining the Acute Angle

Since \(\theta\) must be an acute angle, \(\cos(\theta) = \frac{1}{2} \) corresponds to \(\theta = 60^{\circ}\).

Key Concepts

3D geometry

Three-dimensional (3D) geometry is a branch of mathematics that deals with shapes and figures in a three-dimensional space. In 3D geometry, objects are described not just by length and width, but also by depth. This means they have a spatial quality that helps us understand their position and orientation relative to each other.

Key features and terms in 3D geometry include:

• Points: Basic units in space, identified by coordinates (x, y, z).
• Lines and Planes: Lines extend infinitely in two directions, while planes are flat surfaces extending infinitely in two dimensions.
• Angles and Axis: Important for determining positions and directions of lines or planes.
• Direction Cosines: These are crucial in defining the orientation of lines. They are the cosines of the angles between a line and the coordinate axes.

Understanding these basics is essential for solving complex problems in 3D geometry, like finding angles between lines or between lines and planes.

angles in 3D space

In three-dimensional space, angles help us understand the orientation of lines relative to the coordinate axes. To specify the direction of a line in 3D, we use angles made by the line with the positive directions of the x, y, and z axes.

These angles are known as direction angles, and the cosines of these angles are the direction cosines. If a line makes an angle \( \alpha \) with the x-axis, \( \beta \) with the y-axis, and \( \theta \) with the z-axis, then:

• \(\cos(\alpha)\) for the x-axis, represents the x-component of the line's direction.
• \(\cos(\beta)\) for the y-axis, corresponds to the y-component.
• \(\cos(\theta)\) for the z-axis, is for the z-component.

The sum of the squares of the direction cosines must always equal one:\[\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\theta) = 1\]This equation is vital for determining unknown angles when some angles are provided. It is a reflection of the Pythagorean theorem being applied in 3D space.

acute angle calculation

Understanding how to calculate acute angles is important in many mathematical and practical applications. An acute angle is any angle less than 90 degrees. To find such an angle in the context of 3D geometry and direction cosines, a strategic use of trigonometric identities and properties is necessary.

In the problem where we have the angles that a line makes with two of the axes, we employ:

• Direction Cosine Equation: Use \( \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\theta) = 1 \) to find the missing direction cosine.
• Solve for \( \cos(\theta) \): Knowing this will lead directly to the angle \( \theta \).
• Identifying the Acute Angle: Ensure it is positive and less than 90 degrees, which typically means taking the positive cosine value when options exist.

From the example, once you solve for \(\cos(\theta)\), ensuring it matches the conditions of an acute angle helps confirm the angle. Here, \( \cos(\theta) = \frac{1}{2} \) leads us to the acute angle \( 60^{\circ} \).