## Can you come up with an inequality that holds true for any pair of vectors and their sum? if so, write the inequality and justify

Question

Can you come up with an inequality that holds true for any pair of vectors and their sum? if so, write the inequality and justify your answer.

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41 seconds 2023-01-24T15:40:54+00:00 2 Answers 0 views 0

1. In mathematics, an inequality depicts the relationship between two non-equal values in an algebraic expression.
For any vectors u and v, ||u + v|| ≤ ||u|| + ||v||
By using Cauchy-Schwarz Inequality, we get
u⋅v ≤ ∥u∥⋅∥v∥
∥u + v∥ ≤ ∥u∥ + ∥v∥

### What is meant by inequality?

In mathematics, an inequality depicts the relationship between two non-equal values in an algebraic expression. Inequality signs can indicate that one of the two variables is greater than, greater than or equal to, less than, or less than or equal to another value.
• The triangular inequality means that norm obeys the idea that the shortest distance between two points is a straight line.
• If you go directly from x to y you “travel” ∥x − y∥.
• If you stop at point Tz in between, you travel ∥x − z∥ + ∥z − y∥.
The triangle inequality guarantees that,
∥x − y∥ ≤ ∥x − z∥ + ∥z − y∥ .
For any vectors u and v, ||u + v|| ≤ ||u|| + ||v||
By using Cauchy-Schwarz Inequality, we get
u⋅v ≤ ∥u∥⋅∥v∥
∥u + v∥ ≤ ∥u∥ + ∥v∥