Maximizing and Minimizing Norm of Vector v - w: A Geometric Explanation

The largest value for ||v - w|| would be when the two vectors are pointing in the same direction and the smallest value would be when they are pointing in opposite directions. This can be explained using the triangle inequality. In summary, the largest value for ||v - w|| is 5 and the smallest value is 1, and this can be determined by considering the direction of the vectors.
  • #1
shane1
7
0
I don't know if I'm just having a slow day or what is going on but I am being stumped by this:

If ||v|| = 2 and ||w|| = 3 what are the largest and smallest values possible for ||v - w||. Give a geometric explanation.

Would it be as simple as just adding the two values for the largest, and subtracting for the smallest?

Any help would be apreciated.

-Shane
 
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  • #2
consider the direction, and it is basically the triangle inequality.
 
  • #3
shane1 said:
I don't know if I'm just having a slow day or what is going on but I am being stumped by this:

If ||v|| = 2 and ||w|| = 3 what are the largest and smallest values possible for ||v - w||. Give a geometric explanation.

Would it be as simple as just adding the two values for the largest, and subtracting for the smallest?

Any help would be apreciated.

-Shane

Think about the direction of the two vectors.
 

1. What is meant by "maximizing and minimizing norm of vector v - w"?

The norm of a vector is a measure of its length or magnitude. In this context, "maximizing and minimizing norm of vector v - w" refers to finding the maximum and minimum possible values for the norm of the vector obtained by subtracting vector w from vector v.

2. Why is it important to understand the concept of maximizing and minimizing norm of vector v - w?

Understanding how to maximize and minimize the norm of a vector can be useful in various fields such as mathematics, physics, and engineering. It allows for better understanding of vector operations and can help in solving optimization problems.

3. How is geometric explanation related to maximizing and minimizing norm of vector v - w?

In a geometric explanation, the vectors v and w are represented as arrows in a coordinate system. The difference vector v - w is then drawn from the tip of vector v to the tip of vector w. The length of this difference vector represents the norm of vector v - w, and the direction of the vector gives information about its components. This visual representation helps in understanding the concept of maximizing and minimizing the norm of vector v - w.

4. What are some real-life applications of maximizing and minimizing norm of vector v - w?

The concept of maximizing and minimizing norm of vector v - w has various real-life applications such as in finding the shortest distance between two points, optimizing resource allocation in businesses, and analyzing the performance of athletes in sports.

5. Can the norm of vector v - w ever be negative?

No, the norm of a vector is always a positive value. It represents the distance between the origin and the tip of the vector, and distance is always a positive value. Therefore, the norm of vector v - w can never be negative.

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