Minimum and maximum resultant of three vectors.

In summary, the conversation discusses finding the minimum and maximum resultant of three vectors with given magnitudes. The maximum resultant is found by adding the magnitudes while assuming they are acting in the same direction. The minimum resultant is found by arranging the vectors parallel or antiparallel to each other, with the minimum occurring when they are facing in opposite directions. The conversation also suggests finding the minimum resultant by considering the angle between the vectors and expressing it as a function.
  • #1
Vatsal Goyal
51
6

Homework Statement


In order to solve a question, I need to find the minimum and maximum resultant of three vectors, their magnitudes are given to me.

Homework Equations


Magnitude of vector A = 1
Magnitude of vector B = 3
Magnitude of vector A = 5

The Attempt at a Solution


The maximum part was easy, I just assumed them too be acting in the same direction and added the magnitudes to get the magnitude of the resultant(9 in this case).
But I am not sure about the minimum part. As it cannot form a triangle, it can't be zero. I am guessing that I have to assume them to be arranged parallel or antiparallel to each other in such a way that I get the minimum answer(5-3-1 = 1, in this case). Am I correct? If I am, then is there a strong reason for my answer to be true.
 
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  • #2
Lets start with like you said. 5 pointing one direction(say +x), and 3 pointing exactly in the opposite direction(-x direction), which gives net +2 in the +x direction. Now let's start with the 1 vector starting at the end of that resultant, and being able to spin at any angle. What happens if it is at a 90° angle?
What is the magnitude of the resultant. Perhaps express it as a function of the angle, and see where the minimum occurs.
 
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  • #3
Vatsal Goyal said:
As it cannot form a triangle, ... assume them to be arranged parallel or antiparallel to each other in such a way that I get the minimum answer(5-3-1 = 1, in this case).
Yes.
 
  • #4
scottdave said:
Lets start with like you said. 5 pointing one direction(say +x), and 3 pointing exactly in the opposite direction(-x direction), which gives net +2 in the +x direction. Now let's start with the 1 vector starting at the end of that resultant, and being able to spin at any angle. What happens if it is at a 90° angle?
What is the magnitude of the resultant. Perhaps express it as a function of the angle, and see where the minimum occurs.
Thanks I got it! The minimum would occur when angle between them is 180 degrees meaning it is facing in the -x direction.
 
  • #5
Intuitively, it looks obvious. But if you need to prove it, then that's my approach.
 
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Related to Minimum and maximum resultant of three vectors.

What is the minimum resultant of three vectors?

The minimum resultant of three vectors is the smallest possible magnitude that can be obtained when adding the three vectors together. This occurs when the vectors are arranged in such a way that they cancel each other out, resulting in a magnitude of zero.

What is the maximum resultant of three vectors?

The maximum resultant of three vectors is the largest possible magnitude that can be obtained when adding the three vectors together. This occurs when the vectors are arranged in a way that they all point in the same direction, resulting in the sum of their magnitudes.

How do you calculate the minimum and maximum resultant of three vectors?

The minimum and maximum resultant of three vectors can be calculated using vector addition. The minimum resultant is found by arranging the vectors in a way that they cancel each other out, and then finding the magnitude of the resulting vector. The maximum resultant is found by arranging the vectors in a way that they all point in the same direction, and then adding their magnitudes together.

What factors influence the minimum and maximum resultant of three vectors?

The factors that influence the minimum and maximum resultant of three vectors include the magnitudes and directions of the individual vectors, as well as the angle between them. When arranging the vectors to find the minimum and maximum resultant, the order in which they are added can also affect the outcome.

Why is understanding the minimum and maximum resultant of three vectors important in science?

Understanding the minimum and maximum resultant of three vectors is important in science as it allows us to predict the possible outcomes of vector addition. This is essential in various fields such as physics, engineering, and navigation, where vector operations are used to solve problems and make calculations. It also helps us understand the relationship between different vectors and how they can affect each other when combined.

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