Defining the angle between vectors | Vectors and spaces | Linear Algebra | Khan Academy

TL;DR

Vectors in n dimensions can have angles between them, and we can define the angle using the properties of triangles and the dot product of vectors.

Transcript

A couple of videos ago we introduced the idea of the length of a vector. That equals the length. And this was a neat idea because we're used to the length of things in two- or three-dimensional space, but it becomes very abstract when we get to n dimensions. If this has a hundred components, at least for me, it's hard to visualize a hundred dimensi... Read More

Key Insights

• 🫥 The length of a vector in n dimensions can be calculated using the dot product of the vector with itself.
• 🔺 The triangle inequality ensures that we can always construct a triangle using vectors and their differences.
• 🫥 The dot product of two vectors can be used to calculate the angle between them using the law of cosines.

Questions & Answers

Q: What is the purpose of defining the angle between vectors in higher dimensions?

By defining the angle between vectors in higher dimensions, we are able to apply geometric concepts and relations to these abstract spaces, allowing us to solve problems related to vectors in n-dimensional spaces and perform calculations using the dot product.

Q: What is the triangle inequality and why is it important in this context?

The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the remaining side. In the context of defining angles between vectors, it ensures that we can always construct a triangle using the vectors and their differences.

Q: Can the angle between two vectors ever be undefined?

The angle between two vectors cannot be defined if one or both of the vectors are the 0 vector. This is because the length of the 0 vector is 0, resulting in an undefined value for the cosine of the angle.

Q: What is the difference between perpendicular and orthogonal vectors?

Perpendicular vectors are those that have an angle of 90 degrees between them, while orthogonal vectors are those whose dot product is equal to 0. All perpendicular vectors are orthogonal, but the 0 vector is orthogonal to all vectors, including itself.

Summary & Key Takeaways

• Vectors in n dimensions can be difficult to visualize, but we can still define their lengths and angles using mathematical techniques.

• The length of a vector in n dimensions is a scalar value, which can be calculated using the dot product of the vector with itself.

• To define the angle between two vectors, we can create a triangle using the vectors and their differences, and then apply the law of cosines.