Problem 47 - Master Resource Book in JEE Main Physics

Question

Given, \(\mathbf{A}=4 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}\) and \(\mathbf{B}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\). Which of the following is correct? (a) \(\mathrm{A} \times \mathrm{B}=0\) (b) \(\mathrm{A} \cdot \mathrm{B}=24\) (c) \(\frac{|\mathrm{A}|}{|\mathrm{B}|}=\frac{1}{2}\) (d) \(\mathrm{A}\) and \(\mathrm{B}\) are anti-parallel

Step-by-Step Solution

Answer

Correct: (a) \( \mathrm{A} \times \mathrm{B}=0 \).

Step 1: Understanding the Cross Product

The cross product of two vectors in two dimensions is always zero because the cross product results in a vector perpendicular to the plane defined by the original vectors. In 2D vectors, the cross product results in a scalar zero, hence option (a) is correct.

Step 2: Calculating the Dot Product

The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is calculated as \( \mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y \). For the given vectors: \( 4 \times 2 + 6 \times 3 = 8 + 18 = 26 \). Thus, option (b) is incorrect.

Step 3: Calculating Magnitude and Ratio for Option (c)

The magnitude \( |\mathbf{A}| = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \) and \( |\mathbf{B}| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \). The ratio \( \frac{|\mathbf{A}|}{|\mathbf{B}|} = \frac{\sqrt{52}}{\sqrt{13}} = 2 \). Therefore, option (c) is incorrect.

Step 4: Analyzing the Parallel Condition

Vectors are anti-parallel if they are parallel but point in opposite directions, i.e., one is a negative scalar multiple of the other. Since \( \mathbf{A} \) is 2 times \( \mathbf{B} \), they are parallel, not anti-parallel. Therefore, option (d) is incorrect.

Key Concepts

Understanding the Cross Product

The cross product, also known as the vector product, applies to vectors in three dimensions. However, with two-dimensional vectors, we can still conceptually consider the cross product. In essence, the cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) gives a vector that is perpendicular to the plane containing \( \mathbf{A} \) and \( \mathbf{B} \).
However, since two-dimensional vectors lie on a plane and cannot produce a perpendicular direction in three dimensions, their cross product results in a scalar value of zero. This outcome is significant because it reiterates that 2D vector cross products do not exhibit the same dimensional behavior as 3D vectors.

• The result is zero when dealing with 2D vectors, making option (a) correct in the given exercise.

Calculating the Dot Product

The dot product, sometimes called the scalar product, is a measure of how parallel two vectors are. It results in a scalar and is calculated by multiplying the corresponding components of the two vectors and adding these products together.
For vectors \( \mathbf{A} = 4 \hat{\mathbf{i}} + 6 \hat{\mathbf{j}} \) and \( \mathbf{B} = 2 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}} \), the dot product is:
\[ \mathbf{A} \cdot \mathbf{B} = (4)(2) + (6)(3) = 8 + 18 = 26 \]

• This calculation helps to determine how aligned the vectors are. A larger dot product indicates that the vectors are more aligned, whereas a dot product of zero implies they are perpendicular.
• In the exercise, the expected value was 24, but as calculated, it should be 26, indicating an error in option (b).

Determining Vector Magnitude

Vector magnitude, often referred to as the length or norm of a vector, is a measure of how long the vector is, regardless of direction. It is calculated using the Pythagorean theorem, involving the square root of the sum of the squared components.
For a vector \( \mathbf{A} = 4 \hat{\mathbf{i}} + 6 \hat{\mathbf{j}} \), the magnitude \( |\mathbf{A}| \) is:
\[ |\mathbf{A}| = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \]
Similarly, for \( \mathbf{B} = 2 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}} \):\[ |\mathbf{B}| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

• The magnitude gives a sense of the vector's size without considering direction.
• In the exercise, the ratio \( \frac{|\mathbf{A}|}{|\mathbf{B}|} = \frac{\sqrt{52}}{\sqrt{13}} = 2 \), demonstrating that the expected ratio \( \frac{1}{2} \) in option (c) was incorrect.

This understanding is important for comparing vector lengths in various applications.