Problem 69 - Master Resource Book in JEE Main Physics

Question

Given that \(A\) and \(B\) are greater than 1 . The magnitude of \((\mathbf{A} \times \mathbf{B})\) cannot be (a) equal to \(A B\) (b) less than \(A B\) (c) more than \(A B\) (d) equal to \(A / B\)

Step-by-Step Solution

Answer

The magnitude of \((\mathbf{A} \times \mathbf{B})\) cannot be more than \(AB\).

Step 1: Recall the formula for the cross product magnitude

The magnitude of the cross product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is given by the formula \(|\mathbf{A} \times \mathbf{B}| = AB \sin(\theta)\), where \(\theta\) is the angle between \(\mathbf{A}\) and \(\mathbf{B}\).

Step 2: Analyze the possible value equal to \(AB\)

The magnitude of \(|\mathbf{A} \times \mathbf{B}|\) equals \(AB\sin(\theta)\). For this to be equal to \(AB\), the sine of the angle \(\theta\) must be 1, which implies that \(\theta = 90^\circ\). Hence, \(|\mathbf{A} \times \mathbf{B}| = AB\) is a possible value.

Step 3: Analyze the possibility of being less than \(AB\)

Since \(\sin(\theta)\) ranges from 0 to 1, \(AB\sin(\theta)\) will be less than \(AB\) when \(0 < \theta < 90^\circ\). Therefore, \(|\mathbf{A} \times \mathbf{B}|\) can indeed be less than \(AB\).

Step 4: Analyze the possibility of being more than \(AB\)

For \(|\mathbf{A} \times \mathbf{B}|\) to be more than \(AB\), \(\sin(\theta)\) would need to exceed 1, which is impossible as \(\sin(\theta)\) can only be at most 1. Thus, it can never exceed \(AB\).

Step 5: Examine the possibility of being equal to \(A/B\)

Consider \(A/B\). Since both \(A\) and \(B\) are greater than 1, \(\frac{A}{B}\) is unlikely to match \(AB\sin(\theta)\) for most combinations without additional specific conditions. However, it could take certain values but not broadly feasible like the other options. Yet, this option does not contradict any mathematical constraints like option (c).

Key Concepts

Vector Magnitude

The magnitude is a measure of the length or size of a vector. In physics and mathematics, a vector with a larger magnitude represents a stronger force, a higher speed, or a greater quantity.To find the magnitude of a vector \( \mathbf{A} = (a_1, a_2, a_3) \), you use the formula:\[ |\mathbf{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]This formula calculates the Euclidean norm and gives you a single positive number reflecting the vector's length. When it comes to cross products, the resulting vector also has a magnitude; however, it's a bit more complex.

• For two vectors \( \mathbf{A} \) and \( \mathbf{B} \), their cross product's magnitude is given by \(|\mathbf{A} \times \mathbf{B}| = AB \sin(\theta)\).
• \( \sin(\theta) \) could range from 0 (vectors are parallel) to 1 (vectors are perpendicular).

Understanding vector magnitude involves acknowledging this difference between a vector's magnitude and the magnitude of its cross product with another vector.

Angle Between Vectors

The angle between two vectors is crucial in determining how they interact in space. It is measured using the cosine of that angle. For vectors \( \mathbf{A} \) and \( \mathbf{B} \), you can calculate the angle \( \theta \) between them using their dot product:\[ \cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}||\mathbf{B}|} \]Where:- \( \mathbf{A} \cdot \mathbf{B} \) is the dot product: \( a_1b_1 + a_2b_2 + a_3b_3 \)- \(|\mathbf{A}||\mathbf{B}|\) is the product of their magnitudesThe angle is related to vector operations such as dot products and cross products extensively. When calculating the cross product \(|\mathbf{A} \times \mathbf{B}| = AB \sin(\theta)\):

• If \( \theta = 0^\circ \) or \( 180^\circ \), vectors are parallel, and \( \sin(\theta) = 0 \).
• If \( \theta = 90^\circ \) (perpendicular), \( \sin(\theta) = 1 \), maximizing the product.

This interplay directly affects the magnitude of their cross product and is essential for solving problems involving vector interactions.

Trigonometric Functions in Physics

Trigonometric functions are vital in physics when analyzing wave forms, oscillations, rotations, and many vector operations. They help describe angles and periodic phenomena and are ubiquitous in physics equations.The sine function, in particular, is important for understanding how non-parallel vectors act on one another. As seen with the cross product:\[ |\mathbf{A} \times \mathbf{B}| = AB \sin(\theta)\]

• \( \sin(\theta) \) indicates the projected length of vector \( \mathbf{B} \) in a direction perpendicular to \( \mathbf{A} \).
• As \( \theta \) varies from \( 0\text{ to }90^\circ \), the cross product magnitude changes accordingly.

Beyond vector mathematics, functions like sine and cosine are used to describe oscillations in waves and circular motion, providing frameworks for understanding phenomena such as sound waves, light intensity, and rotational velocities.