Problem 78 - Master Resource Book in JEE Main Physics

Question

The component of a vector \(\mathbf{r}\) along \(X\)-axis will have maximum value if \(\quad\) [NCERT Exemplar] (a) \(\mathrm{r}\) is along positive \(Y\)-axis (b) \(\mathrm{r}\) is along positive \(X\)-axis (c) \(\mathrm{r}\) makes an angle of \(45^{\circ}\) with the \(X\)-axis (d) \(\mathrm{r}\) is along negative \(Y\)-axis

Step-by-Step Solution

Answer

The component of a vector along the X-axis is maximum if the vector is along the positive X-axis (option b).

Step 1: Understanding the Problem

We are asked to find the conditions under which a vector \( \mathbf{r} \) has its maximum component along the \( X \)-axis. This means we need to maximize the projection of \( \mathbf{r} \) on the \( X \)-axis.

Step 2: Conceptual Background

The component of a vector \( \mathbf{r} \) along the \( X \)-axis is given by \( r_x = r \cos(\theta) \), where \( \theta \) is the angle between \( \mathbf{r} \) and the \( X \)-axis. The cosine function, \( \cos(\theta) \), has its maximum value of 1 when \( \theta = 0^\circ \).

Step 3: Analyzing Each Option

Let's evaluate each option:- (a) \( \mathbf{r} \) is along positive \( Y \)-axis: \( \theta = 90^\circ \), \( \cos(90^\circ) = 0 \).- (b) \( \mathbf{r} \) is along positive \( X \)-axis: \( \theta = 0^\circ \), \( \cos(0^\circ) = 1 \).- (c) \( \mathbf{r} \) makes an angle of \( 45^\circ \): \( \cos(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.707 \).- (d) \( \mathbf{r} \) is along negative \( Y \)-axis: \( \theta = 90^\circ \), \( \cos(90^\circ) = 0 \).

Step 4: Identifying the Maximum Component

Among the options, the maximum value of \( \cos(\theta) = 1 \) occurs in option (b) when \( \mathbf{r} \) is along the positive \( X \)-axis. This ensures the component \( r_x = r \) is at its maximum.

Key Concepts

Vector Components

In physics, understanding vectors is essential because they help us describe quantities that have both a magnitude and a direction, such as force, velocity, and displacement. A vector component refers to the projection of a vector onto an axis, which in simpler terms, is how much of the vector aligns with a given direction.
For instance, consider a vector \( \mathbf{r} \) in a two-dimensional plane. This vector can be decomposed into two perpendicular components, one along the x-axis and the other along the y-axis. These components can be visualized as the shadows cast by the vector when the light shines along the other axis.
The component along the x-axis, often denoted as \( r_x \), is calculated by multiplying the magnitude of the vector \( r \) by the cosine of the angle \( \theta \) between the vector and the x-axis: \( r_x = r \cos(\theta) \). This formula comes from trigonometric principles and provides a mathematical view of how vectors split across axes.

• X-component: Shows how much of the vector's effect or movement is in the horizontal direction.
• Y-component: Reveals the vector's influence in the vertical direction.

Understanding vector components is crucial because it allows physicists and engineers to analyze the effects of vectors in individual directions and solve problems involving multiple vectors more easily.

Vector Projection

Vector projection is another fascinating concept that helps us see how one vector can be translated into another's direction. It can be thought of as "dropping" one vector onto the line defined by another vector to measure the extent of its influence in that direction.
For a vector \( \mathbf{a} \) and \( \mathbf{b} \), the projection of \( \mathbf{a} \) onto \( \mathbf{b} \) is given by the formula: \[ \text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b}\]where \( \mathbf{a} \cdot \mathbf{b} \) is the dot product of the vectors, and \( \mathbf{b} \cdot \mathbf{b} \) is the magnitude squared of \( \mathbf{b} \).
Here's why vector projection is useful:

• Applications: It simplifies calculations, such as determining work done by a force, where only the component of force in the direction of motion contributes.
• Visual interpretation: Helps in understanding how much of one vector is 'in' the direction of another vector.

Overall, vector projection is a powerful tool in physics for resolving and comparing vector quantities effectively and efficiently.

Angle Between Vectors

The angle between vectors is a critical measurement that informs us about the directional relationship between two vectors. This angle can reveal how much two vectors are working together or opposing each other.
To find the angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \), use the dot product formula:\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta)\]Rearranging this formula gives the cosine of the angle: \[ \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}\]where \( \theta \) is the angle in radians or degrees.

• Parallel vectors: When \( \cos(\theta) = 1 \), the vectors point in exactly the same direction.
• Perpendicular vectors: If \( \cos(\theta) = 0 \), the vectors are orthogonal, meaning they are at right angles to each other and do not influence each other's directions.
• Opposing vectors: When \( \cos(\theta) = -1 \), the vectors are opposite, indicating a 180° angle between them.

This concept has wide-ranging applications, from physics problems involving forces to computer graphics where determining angles between directions or surfaces is crucial.