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Chapter 2: Problem 261

The resultant of two vectors is maximum when they. (A) are at right angles to each other (B) act in opposite direction (C) act in same direction (D) are act \(120^{\circ}\) to each other

Short Answer

Expert verified
The maximum resultant of two vectors is obtained when they act in the same direction. Therefore, the correct answer is (C) act in the same direction.

Step by step solution

01

Understanding the concept of the resultant vector

The resultant vector is the single vector that can replace the effect of two or more vectors acting at the same time. In other words, it represents the combined effect of these vectors.
02

Creating expressions for the resultant vector

Let's consider two vectors, A and B, with magnitudes A and B, respectively. Let the angle between them be θ. Then, the magnitude R of the resultant vector can be found by using the formula: \( R = \sqrt{A^2 + B^2 + 2AB\cos\theta} \) Now, let's apply this formula to each of the given options.
03

Analyzing option (A) – At right angles to each other

When two vectors are at right angles to each other, θ = \(90^{\circ}\). In this case, cosθ = 0. Using the formula, we get: \( R = \sqrt{A^2 + B^2} \)
04

Analyzing option (B) – Act in opposite direction

When two vectors act in opposite directions, θ = \(180^{\circ}\). In this case, cosθ = -1. Using the formula, we get: \( R = \sqrt{A^2 + B^2 - 2AB} \)
05

Analyzing option (C) – Act in the same direction

When two vectors act in the same direction, θ = \(0^{\circ}\). In this case, cosθ = 1. Using the formula, we get: \( R = \sqrt{A^2 + B^2 + 2AB} \)
06

Analyzing option (D) – At \(120^{\circ}\) to each other

When two vectors are at an angle of \(120^{\circ}\) to each other, θ = \(120^{\circ}\). In this case, cosθ = -1/2. Using the formula, we get: \( R = \sqrt{A^2 + B^2 - AB} \)
07

Finding the maximum resultant

Comparing all options, we find that the maximum resultant is obtained when two vectors act in the same direction: \( R = \sqrt{A^2 + B^2 + 2AB} \). Hence, the correct answer is: (C) act in the same direction

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Most popular questions from this chapter

Comprehensions type questions. A particle is moving in a circle of radius \(\mathrm{R}\) with constant speed. The time period of the particle is T Now after time \(\mathrm{t}=(\mathrm{T} / 6)\) Average speed of the particle is (A) \((\pi \mathrm{R} / 6 \mathrm{~T})\) (B) \([(2 \pi R) / 3 \mathrm{~T}]\) (C) \([(2 \pi R) / T]\) (D) \((\mathrm{R} / \mathrm{T})\)

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A particle is moving in a straight line with initial velocity of $200 \mathrm{~ms}^{-1}\( acceleration of the particle is given by \)\mathrm{a}=3 \mathrm{t}^{2}-2 \mathrm{t}$. Find velocity of the particle at 10 second. (A) \(1100 \mathrm{~ms}^{-1}\) (B) \(300 \mathrm{~ms}^{-1}\) (C) \(900 \mathrm{~ms}^{-1}\) (D) \(100 \mathrm{~ms}^{-1}\)

A body starts its motion with zero velocity and its acceleration is $\left(3 \mathrm{~m} / \mathrm{s}^{2}\right)$. Find the distance travelled by it in fifth second. (A) \(15.5 \mathrm{~m}\) (B) \(17.5 \mathrm{~m}\) (C) \(13.5 \mathrm{~m}\) (D) \(14.5 \mathrm{~m}\)

If \(\mathrm{A}^{\rightarrow} \cdot \mathrm{B}^{\rightarrow}=0\) then (A) \(\left|\mathrm{A}^{\rightarrow}\right|\) must be zero (B) \(\mathrm{B}^{\rightarrow} \mid\) must be zero (C) either \(\mathrm{A}^{\rightarrow}=0, \mathrm{~B}^{\rightarrow}=0\) or \(\theta=0\) (D) either \(\mathrm{A}^{\rightarrow}=0, \mathrm{~B}^{\rightarrow}=0\) or \(\theta=(\pi / 2)\)
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