Problem 22 - Master Resource Book in JEE Main Physics

Question

If the resultant of \(\mathbf{A}\) and \(\mathbf{B}\) makes angle \(\alpha\) with \(\mathbf{A}\) and \(\beta\) with \(\mathbf{B}\), then (a) \(\alpha<\beta\), always (b) \(\alpha<\beta\), if \(AB\) (d) \(\alpha<\beta\), if \(A=B\)

Step-by-Step Solution

Answer

(c) \(\alpha < \beta\), if \(A > B\).

Step 1: Understand the Problem

We are given two vectors \(\mathbf{A}\) and \(\mathbf{B}\), and the angles \(\alpha\) and \(\beta\) are the angles between the resultant vector \(\mathbf{R} = \mathbf{A} + \mathbf{B}\) and the vectors \(\mathbf{A}\) and \(\mathbf{B}\) respectively. We need to determine the condition under which \(\alpha < \beta\).

Step 2: Analyze Geometrically

Visualize the vectors \(\mathbf{A}\) and \(\mathbf{B}\) and their resultant \(\mathbf{R}\). The angle \(\alpha\) is smaller than \(\beta\) if \(\mathbf{R}\) is closer to the direction of \(\mathbf{A}\) than \(\mathbf{B}\). This happens when \(A > B\) because a vector of greater magnitude influences the direction of the resultant more strongly.

Step 3: Use Vector Magnitudes

The magnitude of \(\mathbf{A}\) and \(\mathbf{B}\) influences the direction of \(\mathbf{R}\). If \(A > B\), the vector \(\mathbf{A}\) has a more substantial effect on the direction of \(\mathbf{R}\), making \(\alpha\) smaller. Conversely, if \(B > A\), \(\beta\) would be smaller.

Step 4: Final Conclusion

Thus, \(\alpha < \beta\) when \(A > B\). Therefore, the correct answer is (c).

Key Concepts

Resultant Vector

A resultant vector is what you get when two or more vectors are combined together. Imagine you are walking east, and then you head north. Your overall path isn’t strictly east or north, but a diagonal direction northeast. That's your resultant vector! It embodies both individual movements into one single vector that has a unique direction and length.

In geometry, when two vectors, like \( \mathbf{A} \) and \( \mathbf{B} \), are added together, we use the parallelogram rule or the triangle method to find their resultant vector \( \mathbf{R} \). In our problem, we mainly focus on how this resultant interacts with the individual vectors in terms of angles and direction.

• **A key aspect of resultant vector**: It's a boiled-down representation of combined vector actions.
• **Determines motion**: It shows the effective direction and magnitude when multiple actions (vectors) are involved.

Understanding the resultant vector is crucial because it tells you not just where you’re heading but how strongly you’re heading in that direction compared to the other directions involved.

Angle Between Vectors

The angle between vectors is all about how two vectors "talk" to each other directionally. When looking at the resultant vector \( \mathbf{R} \), the angles \( \alpha \) and \( \beta \) are formed between this resultant and the original vectors \( \mathbf{A} \) and \( \mathbf{B} \) respectively.

These angles are important because they tell you which vector \( \mathbf{A} \) or \( \mathbf{B} \) has more influence on the resultant \( \mathbf{R} \).

• When \( \alpha < \beta \), vector \( \mathbf{A} \) steers the resultant more sharply than vector \( \mathbf{B} \).
• If \( \beta < \alpha \), it’s the reverse, meaning \( \mathbf{B} \) is dominating the direction.

The angle between vectors contributes to understanding how vectors interact spatially, shaping the whole picture of resultant directions.

Vector Magnitude

Vector magnitude is all about the size or length of the vector. Picture it as how far you would travel or how strong the force is, depending on the context – like the horsepower in a car. The magnitude is usually denoted by the length of the arrow representing the vector in diagrams.

When dealing with vectors \( \mathbf{A} \) and \( \mathbf{B} \), their magnitudes help determine which vector is more influential in setting the direction of the resultant vector \( \mathbf{R} \).

• A larger magnitude means a stronger influence on the resultant direction. So, if \( A > B \), \( \mathbf{A} \) will pull \( \mathbf{R} \) closer to its direction, making \( \alpha \) smaller than \( \beta \).
• If \( B > A \), \( \mathbf{B} \) would have more pull, making \( \beta \) smaller than \( \alpha \).

Understanding vector magnitude is essential for predicting and explaining how vectors combine to influence their resultant. It quantifies the strength and defines how much a vector can direct the total vector's outcome.