Triangle Inequality Proven: Prove la + bl ≤ lal + lbl

In summary, the question is asking to deduce the triangle inequality from the given inequality. The solution involves using the property |a|^2 = a dot a and the identity (a+b)dot(a+b) = |a+b|^2.
  • #1
frap
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Homework Statement


I don't know if I'm posting in the right area, but here is the question:
From the inequality: la dot bl= lal lbl lcos(theta)l is less than or equal to lal lbl

Deduce the triangle inequality:
la + bl is less than or equal to lal +lbl


Homework Equations





The Attempt at a Solution



I am not even sure where to begin :(
I'm wondering if I could use numbers to prove it, but that doesn't seem right.
 
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  • #2
Sorry, misread. Think about (|a| + |b|)^2. You also have a property available to you that says |a|^2 = a dot a.
 
Last edited:
  • #3
|a+b|^2 =(a+b)dot(a+b) is a very nice identity application.
 

1. 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 be greater than the length of the third side. In mathematical notation, this can be written as a + b > c, where a, b, and c are the lengths of the three sides of the triangle.

2. How is the Triangle Inequality Theorem proven?

The Triangle Inequality Theorem can be proven using the Law of Cosines, which states that in a triangle with sides a, b, and c and opposite angles A, B, and C, the following equation holds: c^2 = a^2 + b^2 - 2ab cos(C). By rearranging this equation, we can see that c < a + b for all values of cos(C). This proves the Triangle Inequality Theorem.

3. What is the significance of the Triangle Inequality Theorem?

The Triangle Inequality Theorem is important because it helps us determine if a given set of side lengths can form a valid triangle. If the theorem is not satisfied, then the figure is not a triangle. It is also used in many other mathematical proofs and applications, such as in geometry, trigonometry, and optimization problems.

4. How does the Triangle Inequality Theorem relate to real-life situations?

The Triangle Inequality Theorem has many real-life applications, particularly in fields such as engineering and architecture. It helps determine the maximum distance that can be covered by a beam or bridge, and it is also used in navigation systems to calculate the shortest distance between two points on a map.

5. What is the difference between the Triangle Inequality Theorem and the Triangle Inequality Proven?

The Triangle Inequality Theorem is a general mathematical statement that applies to all triangles. The Triangle Inequality Proven is a specific equation that is derived from the Law of Cosines and is used to prove the Triangle Inequality Theorem. Essentially, the Triangle Inequality Proven is a tool that can be used to verify the validity of the Triangle Inequality Theorem in a mathematical proof.

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