Problem 8 - Master Resource Book in JEE Main Physics

Question

The magnitude of the \(X\) and \(Y\) components of \(\mathbf{A}\) are 7 and 6. Also the magnitudes of \(X\) and \(Y\) components of \(\mathbf{A}+\mathbf{B}\) are 11 and 9 respectively. What is the magnitude of \(\mathbf{B}\) ? (a) 5 (b) 6 (c) 8 (d) 9

Step-by-Step Solution

Answer

The magnitude of \( \mathbf{B} \) is 5, so the answer is (a).

Step 1: Understanding the Components

We are given the components of two vectors. For vector \( \mathbf{A} \), the \( X \) and \( Y \) components are 7 and 6, respectively. The vector sum \( \mathbf{A} + \mathbf{B} \) has \( X \) and \( Y \) components of 11 and 9, respectively. Our goal is to find the magnitude of vector \( \mathbf{B} \).

Step 2: Finding Components of Vector B

We know \( \mathbf{A} + \mathbf{B} = (11, 9) \) and \( \mathbf{A} = (7, 6) \). To find the components of \( \mathbf{B} \), we subtract the components of \( \mathbf{A} \) from \( \mathbf{A} + \mathbf{B} \). This gives \( \mathbf{B}_x = 11 - 7 = 4 \) and \( \mathbf{B}_y = 9 - 6 = 3 \).

Step 3: Calculating the Magnitude of Vector B

The magnitude of vector \( \mathbf{B} \) can be calculated using the Pythagorean theorem: \( |\mathbf{B}| = \sqrt{(\mathbf{B}_x)^2 + (\mathbf{B}_y)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \).

Step 4: Identifying the Correct Choice

Based on our calculations, the magnitude of \( \mathbf{B} \) is 5. By comparing with the given answer choices, the correct choice is (a) 5.

Key Concepts

Vector Components

Vectors are fundamental in physics and engineering to describe quantities having both magnitude and direction. When dealing with vectors, it's important to understand how they can be broken down into their components. Every vector in two-dimensional space can be decomposed into two perpendicular components: the X component (horizontal) and the Y component (vertical).
These components are essentially projections of the vector onto the X and Y axes. For example, if we have a vector \( \mathbf{A} \) with components \( (A_x, A_y) \), it means that \( A_x \) represents the horizontal distance, and \( A_y \) represents the vertical distance. These components are foundational because they simplify many vector operations.
Understanding components helps in visualizing how vectors interact. For instance, to find the vector sum \( \mathbf{A} + \mathbf{B} \), we simply add their respective components: \( (A_x + B_x, A_y + B_y) \). This is exactly what was done to determine the components of vector \( \mathbf{B} \) in our problem.

Magnitude Calculation

Once the components of a vector are known, there's an important next step: calculating its magnitude. The magnitude of a vector represents its length and is always a positive value. It tells us the size of the vector without considering its direction.
The formula to calculate the magnitude of a vector \( \mathbf{V} \) with components \( V_x \) and \( V_y \) is \( |\mathbf{V}| = \sqrt{V_x^2 + V_y^2} \). The square root ensures that even if the vector components are negative, the magnitude remains positive, akin to finding the hypotenuse of a right triangle.
For our vector \( \mathbf{B} \), with components \( B_x = 4 \) and \( B_y = 3 \), the magnitude calculation becomes \( |\mathbf{B}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} \), which simplifies to 5. This concise calculation reveals the overall "size" of the vector.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, and it finds a natural application in vector mathematics, especially in magnitude calculation. This theorem states that in a right triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides.
In the context of vectors, each vector can be visualized as the hypotenuse of a right triangle where its components are the other two sides. The relation from the Pythagorean theorem \((c^2 = a^2 + b^2)\) allows us to easily compute the magnitude of a vector also seen as the hypotenuse \(c\) with \(a\) and \(b\) being the X and Y components respectively.
This method of magnitude calculation ensures correctness because it is rooted in a well-established geometrical principle. Applying this to our problem with vector \( \mathbf{B} \), we applied the theorem: \( |\mathbf{B}| = \sqrt{B_x^2 + B_y^2} \). This combines a reliable logical approach with practical calculation, leading us to the solution.