Problem 76 - Master Resource Book in JEE Main Physics

Question

The \(X\) and \(Y\) components of vector A have numerical values 6 and 6 respectively and that of \((\mathbf{A}+\mathbf{B})\) have numerical values 10 and \(9 .\) What is the numerical value of \(\mathbf{B} ?\) (a) 2 (b) 3 (c) 4 (d) 5

Step-by-Step Solution

Answer

(d) 5

Step 1: Understand the Problem

We need to find the vector \( \mathbf{B} \) such that when added to vector \( \mathbf{A} \), the resulting vector \( (\mathbf{A} + \mathbf{B}) \) has components 10 (in the \(X\) direction) and 9 (in the \(Y\) direction). Given \( \mathbf{A} \) has components (6, 6).

Step 2: Find the X Component of Vector B

The \(X\) component of \( \mathbf{A} + \mathbf{B} \) is 10. Since the \(X\) component of \( \mathbf{A} \) is 6, use the equation \( 6 + B_x = 10 \). Solve this by subtracting 6 from both sides: \( B_x = 10 - 6 = 4 \).

Step 3: Find the Y Component of Vector B

The \(Y\) component of \( \mathbf{A} + \mathbf{B} \) is 9. Since the \(Y\) component of \( \mathbf{A} \) is 6, use the equation \( 6 + B_y = 9 \). Solve this by subtracting 6 from both sides: \( B_y = 9 - 6 = 3 \).

Step 4: Calculate the Numerical Value of Vector B

The components of vector \( \mathbf{B} \) are \( B_x = 4 \) and \( B_y = 3 \). The numerical value (magnitude) of \( \mathbf{B} \) is given by \( \sqrt{B_x^2 + B_y^2} \). Substitute in the values: \[ \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]. Therefore, the magnitude of vector \( \mathbf{B} \) is 5.

Key Concepts

Vector Components

In the realm of vector analysis, understanding vector components is crucial. Each vector in a two-dimensional plane can be broken down into two parts, known as its components. These are often referred to as the horizontal (X-component) and the vertical (Y-component). The power of vectors lies within these components; they allow us to analyze and perform calculations more easily.
For the vector \( \mathbf{A} \), given in the exercise:

• X-component: 6
• Y-component: 6

Similarly, the resulting vector \( \mathbf{A} + \mathbf{B} \) has:

• X-component: 10
• Y-component: 9

This information helps you find the components of a mystery vector \( \mathbf{B} \). The components essentially act like coordinates that pinpoint the vector's influence in each direction. By knowing them, you can better understand the vector's placement and direction in space.

Magnitude of a Vector

To fully grasp a vector's role in any situation, figuring out its magnitude is essential. The magnitude conveys the vector's "length" or "strength" in the space it occupies. It is calculated using the Pythagorean theorem, which involves the components of the vector. For vector \( \mathbf{B} \) from our exercise, its components were calculated as:

• X-component \( B_x = 4 \)
• Y-component \( B_y = 3 \)

To find its magnitude, you need to apply the formula: \[ |\mathbf{B}| = \sqrt{B_x^2 + B_y^2} \] Substituting the component values, you get: \[ |\mathbf{B}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] The magnitude of vector \( \mathbf{B} \) turns out to be 5. This measure gives you a scalar value that helps describe the vector's overall impact, disregarding its direction.

Vector Equations

Vector equations are expressions that help understand the relationship between two or more vectors. They can depict how vectors interact, but you must understand each component involved. In the given exercise, the equation of interest arises from the vector addition: \[ \mathbf{A} + \mathbf{B} = (10, 9) \] This equation is split into components:

• \( X \) equation: \( 6 + B_x = 10 \)
• \( Y \) equation: \( 6 + B_y = 9 \)

By solving these component equations, we determine the missing vector \( \mathbf{B} \)'s components. Each component equation provides a step to reach the desired solution.
This exercise illustrates how understanding vector equations aids in dissecting a problem. It shows how specific operations on components lead to comprehending the bigger picture—an essential competence for physics and engineering challenges. Equipped with this knowledge, you can efficiently manage vectors and their calculations.