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]

$$\vec{a} \cdot \vec{b} < \vec{c} \cdot \vec{b}$$ 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 $$\vec{a} \cdot \vec{b} = |\vec{a}| \cdot |\vec{b}| \mathrm{cos(\alpha)}$$

 

[blue_raver22]

The dot product inequality of three vectors a, b, and c in R^d can be visualized geometrically in a few different ways. One way is to think of the dot product as measuring the angle between two vectors. In this case, a.b represents the angle between vectors a and b, while c.b represents the angle between vectors c and b. So, if a.b < c.b, it means that the angle between a and b is smaller than the angle between c and b.

Another way to visualize this inequality is to think of the dot product as a projection of one vector onto another. In this case, a.b represents the projection of vector a onto vector b, while c.b represents the projection of vector c onto vector b. So, if a.b < c.b, it means that the projection of a onto b is smaller than the projection of c onto b.

We can also visualize this inequality in terms of the length of the vectors. The dot product of two vectors can also be calculated as the product of the lengths of the vectors and the cosine of the angle between them. So, if a.b < c.b, it means that the product of the lengths of a and b is smaller than the product of the lengths of c and b.

Overall, the dot product inequality of a, b, and c in R^d can be thought of as a comparison of the angles, projections, and lengths of these vectors. It is important to note that the dot product can also be negative, which would change the direction of the inequality. So, it is important to consider the sign of the dot product when visualizing this inequality.