When does equality hold in the strong triangle inequality Th

When does equality hold in the strong triangle inequality That is, for which rational numbers x and y is | x + y | _2 = max ( |x| _ 2, | y| _ 2)

SOLUTION:

FOR REAL NUMBERS

Let X and Ybe vectors. Then the triangle inequality is given by

|x+y||x|+|y|

(|x+y|)2=(|x|+|y|)2

(x+y)2=(|x|+|y|)2

x2+2xy+y2=|x|2+2|x||y|+|y|2

(|x+y|)2=(|x|+|y|)2

(x+y)2=(|x|+|y|)2

x2+2xy+y2=|x|2+2|x||y|+|y|2

since x2=|x|2x2=|x|2

2xy=2|x||y|xy=|x||y|

And also in other scenario

xy 0 x, y 0 x,y 0

CONCLUSION:

So equality Strongly holds if x and y have the same sign or at least one of them is equal to 0 thus

Equality strongly holds in triangle inequality if both numbers are positive, both are negative or one is zero