Waste less time on Facebook — follow Brilliant.
×

Vector Proof of Cauchy-Schawarz Inequality

Let \(a_1,a_2,a_3,b_1,b_2,b_3\) be six real numbers also consider two vectors \(\vec{A}\) and \(\vec{B}\) such that:-

\[\vec{A}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\] \[\vec{B}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\]

Now let \(\theta\) be the angle between these vectors we know that:- \[\vec{A}.\vec{B}=|\vec{A}||\vec{B}|Cos\theta\] Rearranging this equation gives:- \[Cos\theta = \frac{\vec{A}.\vec{B}}{|\vec{A}||\vec{B}|}\] Putting the values of \(\vec{A}\) and \(\vec{B}\) and applying rules of vector algebra ,equation become:-

\[Cos\theta=\frac{(a_1b_1+a_2b_2+a_3b_3)}{(\sqrt{a_1^2+a_2^2+a_3^2})(\sqrt{b_1^2+b_^2+b_3^2})}\]

Squaring both sides:- \[Cos^2\theta=\frac{(a_1b_1+a_2b_2+a_3b_3)^2}{(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)}\] Now we know that \(Cos^2\theta \leq 1\)........putting the value of \(Cos^2\theta\) in above inequality and rearranging gives cauchy schawarz inequality:- \[(a_1b_1+a_2b_2+a_3b_3)^2 \leq (a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)\]

Note by Aman Sharma
3 years, 2 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...