The angle between two vectors a and b is ∅, where ∅≠90°. Under what conditions will |Proj_{ab}|^{2}+ |Proj_{ba}|^{2}= 1?

Don't really know where to start on this one.

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- Feb 25th 2013, 03:18 PManonymoususer20Projection Vectors Problem
The angle between two vectors a and b is ∅, where ∅≠90°. Under what conditions will |Proj

_{ab}|^{2}+ |Proj_{ba}|^{2}= 1?

Don't really know where to start on this one. - Feb 25th 2013, 03:44 PMILikeSerenaRe: Projection Vectors Problem
Hi anonymoususer20! :)

I'm not entirely sure what you mean with Proj_{ab}.

Presumably it is:

$\displaystyle \text{Proj}_{ab} = \mathbf a \cdot \hat{\mathbf b}$

where $\displaystyle \hat{\mathbf b}$ is the unit vector in the direction of $\displaystyle \mathbf b$.

If that is the case you can substitute $\displaystyle \mathbf a \cdot \hat{\mathbf b} = a \cos \phi$.

And you can substitute something similar for the other projection.

Have you tried that?

If so, what did you get? - Feb 25th 2013, 04:17 PManonymoususer20Re: Projection Vectors Problem
Proj

_{ab}is a projection vector.

|P_{ab}|= |a.b|/|b|. - Feb 25th 2013, 04:21 PMILikeSerenaRe: Projection Vectors Problem
- Feb 25th 2013, 06:58 PMPlatoRe: Projection Vectors Problem

There seems to be somewhat of confusion on notation here.

I have never seen the notation $\displaystyle \text{Proj}_{ab}$.

In North American text books $\displaystyle \text{Proj}_{\vec{b}}\;\vec{a}=\frac{\vec{a}\cdot \vec{b}}{\vec{b}\cdot\vec{b}}\;\vec{b}{$

If we use that definition then $\displaystyle \left\|\frac{\vec{a}\cdot \vec{b}}{\vec{b}\cdot\vec{b}}\;\vec{b}\right\|^2=| \vec{a}\cdot\vec{b}|^2$

Now that makes a really interesting question. - Feb 25th 2013, 10:41 PMILikeSerenaRe: Projection Vectors Problem
- Feb 26th 2013, 04:03 AMPlatoRe: Projection Vectors Problem
- Feb 26th 2013, 04:11 AMILikeSerenaRe: Projection Vectors Problem
$\displaystyle \left\|\frac {\vec{a}\cdot \vec{b}}{\vec{b}\cdot\vec{b}}\;\vec{b} \right\|^2 = \frac{|\vec a \cdot \vec b|^2} {|\vec{b}\cdot\vec{b}|^2} ||\vec b||^2 = \frac{|\vec a \cdot \vec b|^2} {||\vec{b}||^4} ||\vec b||^2 = \frac{|\vec a \cdot \vec b|^2} {||\vec{b}||^2}$