Visualizing the Dot Product Inequality of a, b & c in R^d

[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} &lt; \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)}