Euclidean vector spaces algebra question

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Tallon

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Hello, I require help with the question: What can you say about two nonzero vectors, u and v, that satisfy the equation ||u + v|| = ||u|| + ||v||?

I realise this equation is the triangle inequality for vectors but with both sides equal instead of having the left side less than or equal to the right. It also resembles Pythagoras theorem but without any squares. I just don't know where to begin to come up with an answer to the question.

Thanks in advance!
 
Tallon said:
Hello, I require help with the question: What can you say about two nonzero vectors, u and v, that satisfy the equation ||u + v|| = ||u|| + ||v||?

I realise this equation is the triangle inequality for vectors but with both sides equal instead of having the left side less than or equal to the right. It also resembles Pythagoras theorem but without any squares. I just don't know where to begin to come up with an answer to the question.

Thanks in advance!
Click to expand...
Have you learned to add vectors using "Law of parallelogram"?
 
Tallon said:
Hello, I require help with the question: What can you say about two nonzero vectors, u and v, that satisfy the equation ||u + v|| = ||u|| + ||v||?

I realise this equation is the triangle inequality for vectors but with both sides equal instead of having the left side less than or equal to the right. It also resembles Pythagoras theorem but without any squares. I just don't know where to begin to come up with an answer to the question.

Thanks in advance!
Click to expand...
Where does the triangle inequality come from? Under which conditions does the triangle inequality have equality (it must have equality sometimes else the formula would just have the less than symbol)

If two sides of a triangle have sides of length a and b then what can the length of the 3rd side of the triangle be between?
 
Jomo said:
Where does the triangle inequality come from? Under which conditions does the triangle inequality have equality (it must have equality sometimes else the formula would just have the less than symbol)

If two sides of a triangle have sides of length a and b then what can the length of the 3rd side of the triangle be between?
Click to expand...
From my understanding, the triangle inequality comes from the sum of two sides of a triangle being at least as large as the third. Does that mean for the equation I have asked about, u and v must both not be the longest side of the triangle? Not sure where I am going with this..
 
Tallon said:
Hello, I require help with the question: What can you say about two nonzero vectors, u and v, that satisfy the equation ||u + v|| = ||u|| + ||v||?
Click to expand...
Reminder: \(\displaystyle \|u\|^2=(u\cdot u)\) thus \(\displaystyle \|u+v\|^2=\|u\|^2+2(u\cdot v)+\|v\|^2\)
So \(\displaystyle \left(\|u\|+\|v\|\right)^2=\|u\|^2+2\|u\|\|v\|+\|v\|^2\) so that implies \(\displaystyle 2\|u\|\|v\|=2(u\cdot v)\)
So what can be said?
 
pka said:
Reminder: \(\displaystyle \|u\|^2=(u\cdot u)\) thus \(\displaystyle \|u+v\|^2=\|u\|^2+2(u\cdot v)+\|v\|^2\)
So \(\displaystyle \left(\|u\|+\|v\|\right)^2=\|u\|^2+2\|u\|\|v\|+\|v\|^2\) so that implies \(\displaystyle 2\|u\|\|v\|=2(u\cdot v)\)
So what can be said?
Click to expand...
u and v are equal is the conclusion that I have come to
 
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