# Homework Help: Prove this using vectors

1. Aug 16, 2013

### utkarshakash

1. The problem statement, all variables and given/known data
If a,b,c and k are real constants and α,β,γ are variables subject to the condition that atanα+btanβ+ctanγ = k, then prove using vectors that tan^2 α+tan^2 β+ tan^2 γ ≥ k^2/(a^2+b^2+c^2)

2. Relevant equations

3. The attempt at a solution
$(ai+bj+ck).(tan \alpha i+ tan \beta j+ tan \gamma k) = k \\ k^2 = (a^2+b^2+c^2)(tan^2 \alpha + tan^2 \beta + tan^2 \gamma)$

But what I arrive at is an equation instead of the inequality required to prove.

2. Aug 16, 2013

### ehild

Correct so far...
That is wrong. How is the dot product of two vectors related to their magnitudes?

ehild

3. Aug 16, 2013

### utkarshakash

$\vec{c} ^2 = \vec{c} . \vec{c} = |\vec{c}|^2$

I've used this identity to simplify it further.

4. Aug 16, 2013

### ehild

Yes, but you have the dot product of two different vectors to be squared. $(\vec a \cdot\vec b)^2≠|\vec a |^2 |\vec b|^2$.

ehild

5. Aug 16, 2013

### haruspex

That's only the special case where the two vectors are the same. What is the more general relationship?

6. Aug 16, 2013

### Saitama

Do you know about Cauchy-Schwarz inequality?

7. Aug 16, 2013

### utkarshakash

I encountered it in Calculus but never bothered to go through it as it is not in the JEE syllabus.

8. Aug 16, 2013

### Saitama

Cauchy-Schwarz inequality uses dot product, it isn't too difficult.

Check out Wikipedia.

9. Aug 17, 2013

### utkarshakash

Thanks mate. This inequality helped me to solve some other problems as well.

10. Aug 17, 2013