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

Click For Summary
SUMMARY

The discussion focuses on proving the triangle inequality, specifically the statement that |a + b| ≤ |a| + |b|. The proof utilizes the dot product and properties of norms, particularly the identity |a|^2 = a · a and the expansion of |a + b|^2 = (a + b) · (a + b). Participants emphasize the importance of understanding these mathematical identities to derive the triangle inequality effectively.

PREREQUISITES
  • Understanding of vector dot products
  • Familiarity with norms and absolute values
  • Knowledge of mathematical identities involving vectors
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study vector dot product properties in detail
  • Learn about norms and their applications in geometry
  • Explore mathematical identities related to vector addition
  • Practice proving inequalities in vector spaces
USEFUL FOR

Students in mathematics, particularly those studying linear algebra or vector calculus, as well as educators looking for methods to teach the triangle inequality effectively.

frap
Messages
1
Reaction score
0

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.
 
Physics news on Phys.org
Sorry, misread. Think about (|a| + |b|)^2. You also have a property available to you that says |a|^2 = a dot a.
 
Last edited:
|a+b|^2 =(a+b)dot(a+b) is a very nice identity application.
 

Similar threads

Proving the Triangle Inequality: How to Show llxl-lyll≤lx-yl
  • · Replies 9 ·
Replies
9
Views
6K
Inequality and absolute value proof
  • · Replies 5 ·
Replies
5
Views
5K
How can I prove the statement |a-b| < |a| + |b| using the triangle inequality?
  • · Replies 2 ·
Replies
2
Views
8K
Triangle Inequality: use to prove convergence
  • · Replies 9 ·
Replies
9
Views
2K
Can Absolute Value Inequalities Prove This Expression?
  • · Replies 1 ·
Replies
1
Views
2K
Proving Triangle Inequality: How to Justify Your Work | Homework Tips
  • · Replies 1 ·
Replies
1
Views
2K
Inner Product, Triangle and Cauchy Schwarz Inequalities
  • · Replies 5 ·
Replies
5
Views
4K
How to use the triangle inequality to solve a proof involving absolute values?
  • · Replies 7 ·
Replies
7
Views
3K
Triangle Inequality Proving: Use Sine Law & Find Solution
  • · Replies 2 ·
Replies
2
Views
2K
When does the triangle inequality hold for absolute value?
  • · Replies 5 ·
Replies
5
Views
2K