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

• shybishie
In summary, The geometric interpretation of the equation \vec{a} \cdot \vec{b} < \vec{c} \cdot \vec{b} is that \vec{a} has less of a component in the direction of \vec{b} compared to \vec{c}, or that \vec{a} has a more negative component than \vec{c}. The dot product \vec{a} \cdot \vec{b} can be visualized as a rectangle with sides of length |\vec{a}| and |\vec{b}| \mathrm{cos(\alpha)}, where \alpha is the angle between \vec{a} and \vec{b}. This is because \vec{a
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.

$$\vec{a} \cdot \vec{b} < \vec{c} \cdot \vec{b}$$ 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 $$\vec{c}$$.

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

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

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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.

## 1. What is the dot product inequality in R^d?

The dot product inequality in R^d is a mathematical inequality that relates the dot product of three vectors, a, b, and c, in a d-dimensional space. It states that the absolute value of the dot product of a and b is less than or equal to the product of their magnitudes multiplied by the cosine of the angle between them. In other words, it measures the relationship between the magnitudes of two vectors and the angle between them.

## 2. How is the dot product inequality useful in scientific research?

The dot product inequality is a fundamental concept in linear algebra and has many applications in scientific research. It is commonly used in physics, engineering, and statistics to calculate the angle between two vectors, determine the orthogonality of vectors, and analyze the relationship between variables.

## 3. Can the dot product inequality be visualized in R^d?

Yes, the dot product inequality can be visualized in R^d. In 2-dimensional space, it can be represented geometrically as the angle between two vectors and the length of their projections. In higher dimensions, it can be visualized using vector diagrams or using computer software to plot and manipulate vectors.

## 4. How is the dot product inequality related to the concept of orthogonality?

The dot product inequality is closely related to the concept of orthogonality. When the dot product of two vectors is equal to 0, it means that the vectors are perpendicular or orthogonal to each other. Therefore, the dot product inequality can be used to determine if two vectors are orthogonal based on their magnitudes and the angle between them.

## 5. Can the dot product inequality be extended to higher dimensions?

Yes, the dot product inequality can be extended to any number of dimensions in R^d. The formula remains the same, and it can be applied to vectors in any d-dimensional space. This is what makes the dot product inequality a powerful tool in linear algebra and its applications in various fields of science.

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