Problem 63 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A line \(\mathrm{AB}\) in 3 -dimensional space makes angles \(45^{\circ}\) and \(120^{\circ}\) with the positive \(x\)-axis and the positive \(y\)-axis respectively. If \(A B\) makes an acute angle \(\theta\) with the positive \(z\)-axis, then \(\theta\) equals \([\mathbf{2 0 1 0}]\) (A) \(45^{\circ}\) (B) \(60^{\circ}\) (C) \(75^{\circ}\) (D) \(30^{\circ}\)

Step-by-Step Solution

Answer

\(\theta = 60^{\circ}\)

Step 1: Understand Angles with Axes

Given angles with x and y-axes are 45° and 120° respectively. We need to find the acute angle \( \theta \) with the z-axis.

Step 2: Use Direction Cosines Formula

The direction cosines are represented as \( l, m, n \) with \( \cos(\alpha), \cos(\beta) \) and \( \cos(\theta) \) being the respective cosines of the angles with x, y, and z-axes. Respectively \( \alpha = 45^{\circ} \) and \( \beta = 120^{\circ} \) so we have: \[ l = \cos(45^{\circ}) = \frac{1}{\sqrt{2}}, \ m = \cos(120^{\circ}) = -\frac{1}{2}. \]

Step 3: Apply the Direction Cosines Identity

Use the identity for direction cosines: \( l^2 + m^2 + n^2 = 1 \). Plug in the known values: \[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{2}\right)^2 + n^2 = 1. \]

Step 4: Solve for \(n^2\)

Calculate and solve for \(n^2\): \[ \frac{1}{2} + \frac{1}{4} + n^2 = 1. \] Simplify: \[ \frac{3}{4} + n^2 = 1 \Rightarrow n^2 = \frac{1}{4}. \]

Step 5: Determine \(n\) and \(\theta\)

Since angle \( \theta \) is acute, \( n = \cos(\theta) = \frac{1}{2} \). The angle corresponding to \( \cos(\theta) = \frac{1}{2} \) is \( \theta = 60^{\circ} \).

Key Concepts

3-dimensional space

Three-dimensional space is an area where everything we see in the world around us takes place. Unlike the flat, two-dimensional surfaces, 3D space involves volume because it includes the height, width, and depth of objects.
In our problem, the line \(\overline{AB}\) exists in 3-dimensional space, making angles with the coordinate axes. Each point in this space can be located by a set of numbers \(x, y, z\), representing its position along these three axes.

• The x-axis runs horizontally.
• The y-axis runs vertically.
• The z-axis represents depth, which is what gives space its three-dimensionality.

Understanding 3D space is crucial as it helps in visualizing how objects relate to one another and our surroundings. It's the backbone of mathematics, physics, engineering, and even our everyday navigation.
Using 3-dimensional space principles enables calculations such as finding angles that lines make with these axes, as illustrated in the task where we determine the angle a line makes with the z-axis.

angles with axes

Angles with axes are angles that a line or vector forms with the coordinate axes in a 3-dimensional space.
These angles are important in understanding the direction of lines and vectors within this space. In our example, the line \(\overline{AB}\) makes specific angles with the x-axis and y-axis:

• It makes a \(45\degree\) angle with the x-axis.
• It makes a \(120\degree\) angle with the y-axis.

These angles give us information necessary to find the acute angle \(\theta\) that the line forms with the z-axis. Each of these angles corresponds to a direction cosine, which are the cosines of these angles:

• The direction cosine with the x-axis is \(l = \cos(45\degree) = \frac{1}{\sqrt{2}}\).
• With the y-axis, it is \(m = \cos(120\degree) = -\frac{1}{2}\).

The goal is to find \(n = \cos(\theta)\), which represents the direction cosine with the z-axis. This involves using these previously calculated direction cosines and applying the direction cosines identity.

mathematical problem-solving

Mathematical problem-solving is a systematic approach used to tackle complex questions, especially when dealing with topics like vectors and angles in 3D space.
One effective method is breaking down the problem into smaller, more manageable steps, as illustrated in this exercise. Here, the problem involves finding an unknown angle \(\theta\):

• Initially, we identify the given angles with the x and y-axes.
• Next, we calculate the direction cosines for these known angles.
• We then apply the direction cosines identity: \(l^2 + m^2 + n^2 = 1\).
• By substituting our known values, \(n^2\) is isolated and solved.
• Finally, we determine \(n\) and find \(\theta\) using the inverse cosine function.

This process highlights the importance of direction cosines in 3-dimensional space and how they simplify finding angles with axes.
By approaching problems logically and methodically, the complexity reduces, making it far easier to reach the correct solution.