When is the norm of the sum of 2 vectors=sum of norms

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In summary: It might, but it might not. There are other cases where the vectors are linearly dependent (not perpendicular) where the equality holds.
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Homework Statement


When is |x+y|=|x|+|y| for arbitrary non-zero vectors x,y∈Rn ie, when does equality hold for the well known inequality |x+y|≤|x|+|y|

Homework Equations


|x|2=<x,x>=Σixi2
|x+y|≤|x|+|y|

The Attempt at a Solution


squaring both sides of the inequality we have
(|x+y|)2≤|x|2+|y|2+2|x||y|,
but (|x+y|)2=∑i(xi+yi)2=∑ixi2+∑iyi2+2∑ixiyi, then, by definition of norm we can cancel out all terms in the inequality but
2∑ixiyi=<x,y>≤2|x||y| so we see that equality can only hold when both sides of this inequality are equal, therefore we square both sides
<x,y>2=|x|2|y|2, by definition this expands to ∑ixiyi⋅∑ixiyi=
i(xiyi)2+∑i≠jxixjyiyj
but ∑i(xiyi)2=|x|2|y|2
so our condition for equality of |x+y|=|x|+|y| simplifies to ∑i≠jxixjyiyj=0 or
i≠jxixjyiyj=0
this is the answer I arrived at, however I don't feel confident in it for some reason. I found other solutions online for the problem in this exact book that state "when x and y are linearly dependent", however in the newest edition of the book, which I have, the problem actually states : "hint: the answer is not 'they are linearly dependent.'"

Edit: I've already found some errors in my work, including the fact that I did not expand (|x||y|)^2 properly
 
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  • #2
my ammended solution is ∑i≠jxiyj-xjyi=0
 
  • #3
cpsinkule said:
my ammended solution is ∑i≠jxiyj-xjyi=0

You are really squaring too much here and drawing wrong conclusions. Think about the dot product. You should get just x.y=|x||y| which you had about halfway through. Then remember that the dot product of two vectors is ##|x||y|cos(\theta)## where ##\theta## is the angle between the two vectors. So?
 
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  • #4
Dick said:
You are really squaring too much here and drawing wrong conclusions. Think about the dot product. You should get just x.y=|x||y| which you had about halfway through. Then remember that the dot product of two vectors is ##|x||y|cos(\theta)## where ##\theta## is the angle between the two vectors. So?
i see that, however the book explicitly states "the answer is NOT that they are linearly dependent" aka θ=0 which is why I am so confused, the book is "Calculus on Manifolds" by Spivak if you were wondering.
 
  • #5
cpsinkule said:
i see that, however the book explicitly states "the answer is NOT that they are linearly dependent" aka θ=0 which is why I am so confused, the book is "Calculus on Manifolds" by Spivak if you were wondering.
"Linearly dependent" doesn't necessarily mean that θ=0, as you said. If the angle between two vectors were 180°, so that they point in opposite directions, the vectors would be linearly dependent, but it does not follow automatically that |x + y| = |x| + |y|.
 

1. What is the norm of a vector?

The norm of a vector is a measure of its length or magnitude. It is calculated by taking the square root of the sum of the squares of each component of the vector.

2. What is the sum of two vectors?

The sum of two vectors is a new vector that results from adding the corresponding components of the two original vectors. For example, if vector A has components (1,2) and vector B has components (3,4), the sum of A and B would be a new vector with components (4,6).

3. When is the norm of the sum of two vectors equal to the sum of their norms?

The norm of the sum of two vectors is only equal to the sum of their norms when the two vectors are orthogonal (perpendicular) to each other. In other words, when the dot product of the two vectors is equal to 0.

4. Is the norm of the sum of two vectors always equal to the sum of their norms?

No, the norm of the sum of two vectors is not always equal to the sum of their norms. This only occurs when the two vectors are orthogonal to each other.

5. How can the concept of norms and vector addition be applied in real-world situations?

The concept of norms and vector addition is widely used in various fields such as physics, engineering, and computer science. For example, in physics, the velocity of an object can be represented as a vector and its magnitude (norm) can be calculated using vector addition. In computer science, vector addition is used in graphics and animation to manipulate and combine multiple vectors to create complex shapes and movements.

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