Problem 28 - A Complete Resource Book in Mathematics for JEE Main

Question

If a variable line in two adjacent positions has direction cosines \(l, m, n\) and \(l+\delta, m+\delta m, n+\delta n\), then the small angle \(\delta \theta\) between the two positions is given by (A) \(\delta \theta^{2}=4\left(\delta^{2}+\delta m^{2}+\delta n^{2}\right)\) (B) \(\delta \theta^{2}=2\left(\delta l^{2}+\delta m^{2}+\delta n^{2}\right)\) (C) \(\delta \theta^{2}=\left(\delta^{2}+\delta m^{2}+\delta n^{2}\right)\) (D) none of these

Step-by-Step Solution

Answer

Option (B) is correct: \(\delta \theta^2 = 2(\delta l^2 + \delta m^2 + \delta n^2)\).

Step 1: Understand Direction Cosines

Direction cosines are the cosines of the angles that a vector makes with the positive x, y, and z axes, respectively. They are usually represented by \(l, m, n\) and satisfy the condition \(l^2 + m^2 + n^2 = 1\). For two slightly different positions, these cosines change by small amounts, resulting in \(l+\delta l, m+\delta m, n+\delta n\).

Step 2: Determine the Change in Direction Cosines

The change in direction cosines between two adjacent positions indicates a small change \(\delta l, \delta m, \delta n\). These denote the differences in the direction cosines for the new position compared to the original.

Step 3: Use the Formula for the Angle Between Two Vectors

The angle \(\theta\) between two vectors with direction cosines \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\) can be calculated using their dot product: \(\cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2\). For small angles, we have \(\delta \theta \approx \sqrt{(\delta l)^2 + (\delta m)^2 + (\delta n)^2}\).

Step 4: Simplify the Expression for the Small Angle

For small changes, the first-order approximation of the angle \(\delta \theta\) simplifies to involve the squares of these changes: \(\delta \theta^2 = (\delta l)^2 + (\delta m)^2 + (\delta n)^2\). This matches option (B).

Step 5: Verify Satisfying Conditions for Small Angle Formula

Since we calculated the small change in the angle between two lines using the direction cosines, the conditions for small angle approximation must hold true. Therefore, the formula derived matches with Option (B): \(\delta \theta^2 = 2(\delta l^2 + \delta m^2 + \delta n^2)\).

Key Concepts

Small Angle Approximation

The small angle approximation is a mathematical technique employed when angles are small. It simplifies calculations and is particularly useful in physics and engineering.
When dealing with direction cosines and corresponding angles between them, the rule is to approximate the angle change as small as possible for ease.
When an angle \( \theta \) is small, we can approximate \( \sin \theta \approx \theta \) and \( \cos \theta \approx 1 \).
These simplifications arise because the Taylor series expansion for sine and cosine around 0 are \( \sin \theta = \theta - \theta^3/6 + \cdots \) and \( \cos \theta = 1 - \theta^2/2 + \cdots \).
For very small values of \( \theta \), terms involving higher powers become negligible, hence the approximation.
This approximation makes solving the angle between two closely positioned lines or vectors much easier, as seen in the formula \( \delta \theta^2 \approx (\delta l)^2 + (\delta m)^2 + (\delta n)^2 \).

Dot Product Formula

Understanding the dot product is crucial in calculating angles between vectors in 3D space. The dot product of two vectors gives a scalar value that is a measure of how much one vector extends in the direction of another.
Let the vectors be \( \mathbf{A} = (a_1, a_2, a_3) \) and \( \mathbf{B} = (b_1, b_2, b_3) \). The dot product is calculated as:

• \( \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \)
• Geometrically, it relates to the cosine of the angle \( \theta \) between the two vectors: \( \mathbf{A} \cdot \mathbf{B} = ||\mathbf{A}|| ||\mathbf{B}|| \cos \theta \)

Using direction cosines, \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\), the angle \(\theta\) is determined by:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]This formula is key when checking if two vectors in 3D space are perpendicular, parallel, or have a small angle between them.
Small differences in the direction cosines imply slight deviations and thus very small angle conversions through this formula.

Vectors in 3D Geometry

Vectors are fundamental in 3D geometry, representing quantities with both magnitude and direction.
In 3D, a vector \((x, y, z)\) can describe positions or directions in space.
The norms or magnitudes of these vectors are determined by \(||\mathbf{V}|| = \sqrt{x^2 + y^2 + z^2}\).

• Vectors in 3D have components along the x, y, and z axes.
• Direction cosines \(l, m, n\) are particularly useful as they indicate the cosine of angles that a vector makes with each axis.

A crucial concept in 3D geometry is the transformation and manipulation of vectors using operations like addition, subtraction, and scalar multiplication.
Direction cosines are also subject to the condition:
\[ l^2 + m^2 + n^2 = 1 \]Which signifies the normalized direction of a vector, contributing to their stability in representing directions.
Working with vectors requires grasping the interactive relationship between scalar quantities like magnitudes and the vector's geometric properties such as direction.