Problem 63 - Master Resource Book in JEE Main Physics

Question

A vector \(\mathbf{F}_{1}\) is along the positive \(Y\)-axis. If its vector product with another vector \(\mathbf{F}_{2}\) is zero, then \(\mathbf{F}_{2}\) could be (a) \(4 \hat{\mathrm{j}}\) (b) \(\hat{\mathrm{j}}+\hat{\mathrm{k}}\) (c) \(\hat{\mathrm{j}}-\hat{\mathrm{k}}\) (d) \(-4 \hat{\mathrm{i}}\)

Step-by-Step Solution

Answer

Option (a) is correct: \(\mathbf{F}_{2} = 4 \hat{\mathbf{j}}\).

Step 1: Understanding the Problem

Given two vectors, we need to find the condition under which their cross product is zero. The cross product of two vectors is zero if they are parallel or if one of them is the zero vector.

Step 2: Identify the Known Vector

The vector \(\mathbf{F}_{1}\) is along the positive \(Y\)-axis. In unit vector notation, this can be expressed as \(\mathbf{F}_{1} = c \hat{\mathbf{j}}\), where \(c\) is a constant.

Step 3: Understand the Zero Cross Product Condition

For \(\mathbf{F}_{1} \times \mathbf{F}_{2} = \mathbf{0}\), \(\mathbf{F}_{2}\) must be parallel to \(\mathbf{F}_{1}\) or zero. Parallel means \(\mathbf{F}_{2}\) has no component outside the \(\hat{\mathbf{j}}\) direction.

Step 4: Consider Each Option

Expanding on each option, we have: 1. **Option (a)**: \(4 \hat{\mathbf{j}}\) is parallel to \(\hat{\mathbf{j}}\), so the cross product is zero.2. **Option (b)**: \(\hat{\mathbf{j}} + \hat{\mathbf{k}}\) has a \(\hat{\mathbf{k}}\) component, which means the cross product is not zero.3. **Option (c)**: \(\hat{\mathbf{j}} - \hat{\mathbf{k}}\) has a \(\hat{\mathbf{k}}\) component, which means the cross product is not zero.4. **Option (d)**: \(-4 \hat{\mathbf{i}}\) is along the \(\hat{\mathbf{i}}\) direction, which is perpendicular to \(\hat{\mathbf{j}}\) and results in a nonzero product.

Step 5: Conclusion

The only vector \(\mathbf{F}_{2}\) that is parallel to \(\hat{\mathbf{j}}\) and makes the cross product zero is \(4 \hat{\mathbf{j}}\).

Key Concepts

Parallel Vectors

When two vectors are parallel, they share the same line of action or lie on the same line even if they point in opposite directions. This concept is crucial in understanding certain vector operations, such as the cross product. In the context of the exercise, we are given a vector \( \mathbf{F}_{1} \) along the positive \( Y \)-axis. If another vector, say \( \mathbf{F}_{2} \), is parallel to \( \mathbf{F}_{1} \), it will either point directly in the \( \hat{\mathbf{j}} \) direction, or directly opposite. This means that there are no components of \( \mathbf{F}_{2} \) in the \( \hat{\mathbf{i}} \) or \( \hat{\mathbf{k}} \) directions.

• Parallel vectors have the form \( \mathbf{F}_{2} = k \mathbf{F}_{1} \), where \( k \) is a scalar.
• If \( \mathbf{F}_{1} = c \hat{\mathbf{j}} \), \( \mathbf{F}_{2} = 4\hat{\mathbf{j}} \) or any multiple of \( \hat{\mathbf{j}} \) is parallel.

Parallel vectors often simplify problems because certain calculations result in zero, such as with the cross product.

Zero Cross Product

The cross product of two vectors, \( \mathbf{A} \times \mathbf{B} \), results in a vector that is perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \). If this product equals zero, it indicates that the vectors are either parallel or one is the zero vector. For a vector cross product to be zero, it's crucial for the vectors to lack any perpendicular components to each other.

• The mathematical expression for a zero cross product is \( \mathbf{A} \times \mathbf{B} = \mathbf{0} \).
• This can only occur if one vector is a scalar multiple of the other, indicating parallelism.
• If one vector is zero, the cross product is trivially zero due to multiplication rules.

Understanding this outcome helps in both theoretical problems and practical applications in physics and engineering.

Vector Components

Vectors in three-dimensional space are usually expressed in terms of their components along the \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \) unit vectors, which represent the \( x, y, \) and \( z \) directions respectively. Each component represents the projection of the vector along the respective axis, and knowing these is essential for vector arithmetic.

• A vector \( \mathbf{F} = a\hat{\mathbf{i}} + b\hat{\mathbf{j}} + c\hat{\mathbf{k}} \) has components along three axes.
• Vectors can be broken down to understand their contributions in each direction, simplifying analysis and calculations.

In the context of the exercise, identifying that \( \mathbf{F}_{1} \) and its potential parallel vectors have no \( \hat{\mathbf{i}} \) or \( \hat{\mathbf{k}} \) components made it clear they solely affected the \( y \)-direction.