How to visualize the dot product inequality a·b < c·b geometrically

[shybishie]

Suppose I have three vectors a,b and c in $$R^d$$, And, I have that a.b < c.b(assume Euclidean inner product). What are the ways to visualize relation between a,b and c geometrically? I realize this is slightly open-ended, but am looking for insight here. Thanks in advance.

PS: I have a thought or two, but I'd like to hear feedback before I give my view of the situation.

 

[markiv]

$$\vec{a} \cdot \vec{b} < \vec{c} \cdot \vec{b}$$ can be interpreted geometrically as $$\vec{a}$$ having less of a component in the direction of $$\vec{b}$$ than does $$\vec{c}$$. Or $$\vec{a}$$ has a more negative component than $$\vec{c}$$.

 

[shybishie]

Thank you, markly. In retrospect, I should have framed this question to be less trivial sounding than it came out.

 

[Edgardo]

The dot product $$\vec{a} \cdot \vec{b}$$ can be visualized as a rectangle (see orange rectangle http://www.ies.co.jp/math/java/vector/naiseki_e/naiseki_e.html" ) having sides of length $$|\vec{a}|$$ and $$|\vec{b}| \mathrm{cos(\alpha)}$$.

This is because $$\vec{a} \cdot \vec{b} = |\vec{a}| \cdot |\vec{b}| \mathrm{cos(\alpha)}$$