Problem 38 - Master Resource Book in JEE Main Physics

Question

Given, \(\mathbf{C}=\mathbf{A} \times \mathbf{B}\) and \(\mathbf{D}=\mathbf{B} \times \mathbf{A}\). What is the angle between \(\mathbf{C}\) and \(\mathbf{D}\) ? [WB JEE 2009] (a) \(30^{\circ}\) (b) \(60^{\circ}\) (c) \(90^{\circ}\) (d) \(180^{\circ}\)

Step-by-Step Solution

Answer

The angle is 180° (option d).

Step 1: Understanding Cross Products

The vector cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) results in a vector \( \mathbf{C} = \mathbf{A} \times \mathbf{B} \) that is perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \). Similarly, the cross product \( \mathbf{D} = \mathbf{B} \times \mathbf{A} \) also results in a vector perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \).

Step 2: Recognizing Vector Directions

The direction of vectors resulting from cross products are determined by the right-hand rule. This means \( \mathbf{C} = \mathbf{A} \times \mathbf{B} \) points in the opposite direction relative to \( \mathbf{D} = \mathbf{B} \times \mathbf{A} \).

Step 3: Angle Between Opposite Directions

Since \( \mathbf{C} \) and \( \mathbf{D} \) are in opposite directions due to the properties of cross products, the angle between them is \( 180^{\circ} \).

Step 4: Solution Summary

The angle between vectors \( \mathbf{C} \) and \( \mathbf{D} \) is opposite, and thus the angle is \( 180^{\circ} \). Therefore, the correct answer is option (d).

Key Concepts

Right-Hand Rule

The right-hand rule is a useful guide for finding the direction of the cross product between two vectors. To use the right-hand rule, you curl the fingers of your right hand from the first vector \( \mathbf{A} \) towards the second vector \( \mathbf{B} \). Then, your thumb points in the direction of the cross product \( \mathbf{C} = \mathbf{A} \times \mathbf{B} \).
This rule is crucial in physics and engineering for determining rotational directions and forces. It helps in visualizing the 3D space orientation and ensures that the resulting vector is oriented correctly relative to the initial two vectors.

• Always start by positioning your hand so the fingers curl from the first vector toward the second.
• Your thumb will naturally point in the direction of the resultant vector.
• Remember that reversing the order of vector multiplication, as in \( \mathbf{B} \times \mathbf{A} \), changes the direction of the resulting vector.

Perpendicular Vectors

In the context of vector cross products, the resulting vector is always perpendicular to the original vectors. For example, if you calculate \( \mathbf{C} = \mathbf{A} \times \mathbf{B} \), \( \mathbf{C} \) is perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \). Geometrically, this perpendicularity forms the basis of many vector calculations, ensuring vectors in different planes are non-parallel.
The property of perpendicular vectors is useful when considering physical systems where two or more forces interact. This characteristic allows solving problems by projecting vectors along axes in 3D geometry.

• The cross product yields a vector that forms a right angle with the original vectors.
• This property is pivotal for various applications, including torque calculation in physics.
• It also helps understand the basic setup of Cartesian coordinate systems where unit vectors like \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) are orthogonal to each other.

Angle Between Vectors

Vectors \( \mathbf{C} \) and \( \mathbf{D} \), resulting from different orders of cross product, are interesting because they highlight a unique case where the angle between them is precisely \( 180^{\circ} \). This occurs because \( \mathbf{C} = \mathbf{A} \times \mathbf{B} \) is the direct opposite of \( \mathbf{D} = \mathbf{B} \times \mathbf{A} \), meaning their directions are antiparallel.
Understanding angles between vectors is fundamental, not only in theoretical math but also in real-world problems that require assessing the direction and magnitude of forces.

• The angle \( 180^{\circ} \) simply confirms that two vectors are pointing in exactly opposite directions.
• Cross products often yield vectors that are just opposite, showcasing the need for careful ordering in calculations.
• Recognizing these properties can lead to simplifications in problem-solving, especially in dynamics and motion studies.