Minimum and maximum resultant of three vectors.

[Vatsal Goyal]

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.

 

[scottdave]

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.

 

[haruspex]

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.

 

[Vatsal Goyal]

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.

 

[scottdave]

Intuitively, it looks obvious. But if you need to prove it, then that's my approach.