Why Is the Discriminant Non-Positive in the Triangle Inequality Proof?

Therefore, the inequality ##D=4(Re(x,y))^2-4 ||y||^2|x||^2 \leq 0## must hold, since a positive discriminant would imply two distinct real solutions for ##t##, which contradicts the fact that ##f(t)## can have at most one solution.
  • #1
LagrangeEuler
717
20
In the derivation of triangle inequality [tex]|(x,y)| \leq ||x|| ||y||[/tex] one use some ##z=x-ty## where ##t## is real number. And then from ##(z,z) \geq 0## one gets quadratic inequality
[tex]||x||^2+||y||^2t^2-2tRe(x,y) \geq 0[/tex]
And from here they said that discriminant of quadratic equation
[tex]D=4(Re(x,y))^2-4 ||y||^2|x||^2 \leq 0 [/tex]
Could you explain me why ##<## sign in discriminant relation? When discriminant is less then zero solutions are complex conjugate numbers. But I still do not understand the discussed inequality. What about for example in case
[tex]||x||^2+||y||^2t^2-2tRe(x,y) \leq 0[/tex]?
 
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  • #2
The quadratic equation
$$
||x||^2+||y||^2t^2-2tRe(x,y) \geq 0
$$
must always be zero or positive for all real ##t##. This means the function ##f(t)=||x||^2+||y||^2t^2-2tRe(x,y)## must always be above the ##t## axis, which further implies that it can have at most one solution (which occurs when the minimum of this function touches the ##t## axis): one solution or no solution at all. In terms of the discrimant, the discrimant of ##f(t)## is either zero or negative.
 

Related to Why Is the Discriminant Non-Positive in the Triangle Inequality Proof?

What is the triangle inequality theorem?

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side.

How is the triangle inequality theorem used in mathematics?

The triangle inequality theorem is used in a variety of mathematical concepts, including geometry, trigonometry, and calculus. It is particularly important in proving geometric theorems, such as the Pythagorean theorem, and in solving optimization problems.

What is the relationship between the triangle inequality theorem and the Law of Cosines?

The Law of Cosines is a generalization of the Pythagorean theorem and can also be derived from the triangle inequality theorem. The Law of Cosines is used to find the length of a side of a triangle when the lengths of the other two sides and the included angle are known.

Can the triangle inequality theorem be applied to any type of triangle?

Yes, the triangle inequality theorem applies to all types of triangles, including acute, right, and obtuse triangles. It is a fundamental property of triangles and holds true for all possible combinations of side lengths.

How can the triangle inequality theorem be used in real-world situations?

The triangle inequality theorem has many practical applications, such as in engineering, architecture, and navigation. It is used to determine the shortest distance between two points, the maximum height of a bridge, and the shortest flight path for airplanes.

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