Understanding the Dot Product of Orthogonal Vectors

In linear algebra and vector calculus, the dot product, or scalar product, is a fundamental operation that provides valuable information about the relationship between two vectors. Specifically, if two vectors are orthogonal, their dot product is a key piece of information that helps us understand their properties. In this article, we delve into the concept of the dot product, focusing on the scenario where two vectors are orthogonal, and explain why the dot product of two orthogonal vectors is always zero.

The Dot Product of Vectors

The dot product of two vectors is a scalar quantity that is defined as the sum of the products of the corresponding entries of the vectors. For two vectors A and B, their dot product is calculated as:

AB A · B A B cos

where:

A and B are the magnitudes (lengths) of the vectors A and B, respectively. θ is the angle between the vectors A and B. cos θ is the cosine of the angle θ between the vectors.

This definition is particularly useful when the vectors are orthogonal, as the cosine of 90 degrees (orthogonal angle) is 0.

The Dot Product of Orthogonal Vectors

If two vectors are orthogonal, the angle between them is 90 degrees. In the case of orthogonal vectors, the dot product is:

AB A · B A B cos 90°

Since cos 90° 0, it follows that:

AB A B * 0 0

Hence, the dot product of orthogonal vectors is always zero, regardless of the magnitudes of the vectors. This property is a direct consequence of the geometric interpretation of the dot product, which measures the projection of one vector onto another.

Visualizing Orthogonal Vectors and Their Dot Product

To better understand this concept, it can be helpful to visualize orthogonal vectors. If vectors A and B are orthogonal, they are perpendicular to each other, and the dot product can be seen as the projection of one vector onto the other. Since the projection of a vector in a direction perpendicular to itself is zero, the dot product of orthogonal vectors is always zero.

Examples of Dot Product Calculations

Let's consider a specific example with orthogonal vectors in a two-dimensional space. Suppose we have two vectors x and y:

x [x1, x2]

y [y1, y2]

The dot product of these vectors is calculated as:

xy x1y1 x2y2

When the vectors are orthogonal, the angle θ between them is 90 degrees, so:

xy x1y1 x2y2 cos 90° 0

This means that the dot product is zero, confirming our theoretical result.

Conclusion

In summary, the dot product of two orthogonal vectors is always zero, as the cosine of the angle between them is zero. This property is crucial in various applications of linear algebra and vector calculus, such as in physics, engineering, and computer science. Understanding this concept helps in analyzing and solving problems related to vector spaces and linear transformations. For a detailed visual understanding, consider watching the video tutorial on Lecture 2b for a more intuitive grasp of the concept.