Problem 36 - Master Resource Book in JEE Main Physics

Question

Assertion A vector A points vertically upwards and \(\mathbf{B}\) points towards north. The vector product \(\mathbf{A} \times \mathbf{B}\) is along east. Reason The direction of \(\mathbf{A} \times \mathbf{B}\) is given by right hand rule.

Step-by-Step Solution

Answer

The assertion and the reason are both true.

Step 1: Understand the vectors

Vector \( \mathbf{A} \) points vertically upwards, which means along the positive \( z \)-axis if we assume the ground is the \( xy \)-plane. Vector \( \mathbf{B} \) points towards the north, which corresponds to the positive \( y \)-axis.

Step 2: Define the axes and vectors

Assign the directions: east (\( x \)-axis), north (\( y \)-axis), and upwards (\( z \)-axis). Thus, \( \mathbf{A} = (0, 0, A) \) and \( \mathbf{B} = (0, B, 0) \), where \( A \) and \( B \) are magnitudes of the vectors \( \mathbf{A} \) and \( \mathbf{B} \), respectively.

Step 3: Calculate the cross product

The cross product \( \mathbf{A} \times \mathbf{B} \) is calculated using the formula for vector cross product: \( (a_1, a_2, a_3) \times (b_1, b_2, b_3) = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \). For \( \mathbf{A} \) and \( \mathbf{B} \), it becomes \( (0 \cdot 0 - A \cdot 0, A \cdot 0 - 0 \cdot B, 0 \cdot B - 0 \cdot 0) = (0, 0, AB) \). Therefore, \( \mathbf{A} \times \mathbf{B} = AB(1, 0, 0) \), which is along the x-axis.

Step 4: Determine the direction of the cross product result

In our coordinate system, the result \( (1, 0, 0) \) means the product is along the positive \( x \)-axis, which is east. Hence, the assertion that \( \mathbf{A} \times \mathbf{B} \) is along east is correct.

Step 5: Verify the Reason with Right-Hand Rule

The right-hand rule confirms the direction of \( \mathbf{A} \times \mathbf{B} \) by physically aligning the right hand: point your fingers along \( \mathbf{A} \) (upwards), and curl them towards \( \mathbf{B} \) (north). Your thumb points east, confirming the direction found in Step 4. Therefore, the reason is also correct.

Key Concepts

Right-Hand Rule

When working with vector cross products, the Right-Hand Rule is a helpful tool for determining the direction of the resulting vector. To use this rule, extend your right hand and align your fingers in the direction of the first vector. In our example, vector \( \mathbf{A} \) points vertically upwards. Then curl your fingers towards the direction of the second vector, which is towards the north for vector \( \mathbf{B} \). Your thumb will naturally point towards the resulting direction of the cross product, ensuring the direction is understood intuitively.

This physical gesture quickly verifies the direction without complex calculations. It's important to remember that this rule only applies in three-dimensional space, as vector cross products do not exist in two dimensions. For other geometrical interpretations or larger dimensions, other rules or methods may be applied.

Coordinate System Axes

In understanding the vector cross product, it's crucial to know the coordinate system you're working within. In our example, the convention used is the Cartesian coordinate system. The axes are defined as follow:
- The positive \( x \)-axis represents 'east'.
- The positive \( y \)-axis represents 'north'.
- The positive \( z \)-axis represents 'upwards'.

Establishing this coordinate setup is key before proceeding with calculations, as it determines the representation of each vector's direction. This representation simplifies the step towards calculating the cross product, allowing us to label vectors such as \( \mathbf{A} = (0, 0, A) \) and \( \mathbf{B} = (0, B, 0) \), corresponding to their respective axes. Once the vectors are aligned with the axes, applying the right-hand rule or other calculations becomes much clearer and intuitive.

Direction of Vectors

The direction of vectors is foundational in calculating their cross product. Initially, we identify our vectors' directions in the coordinate system. For vectors \( \mathbf{A} \) and \( \mathbf{B} \), \( \mathbf{A} \) is vertically upwards aligned with the \( z \)-axis, and \( \mathbf{B} \) points north along the \( y \)-axis.

When performing the cross product \( \mathbf{A} \times \mathbf{B} \), the resulting vector will be perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \). In three-dimensional space, this often means the resulting direction can be found using a combined understanding of the Right-Hand Rule and coordinate setup.

After calculation, we find that the resultant vector aligns with the vector along the east or \( x \)-axis direction. This conclusion is only valid once the direction of the original vectors within the coordinate system is clearly understood, making direction a critical component of vector cross products.