Unit vectors | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

Unit vectors are vectors with a length of 1, and they are useful for transformations and projections in vector spaces.

Transcript

We covered the idea of vector length many, many videos ago. And I realize that I forgot to cover an important topic. This topic's going to be useful when we do some types of transformation -- actually, the projections that I'll do in the next video. The notion that I forgot to do is the notion of a unit vector. And all this is, is a vector that has... Read More

Key Insights

• 👾 A unit vector has a length of 1 and is used to represent direction in vector spaces.
• 🇦🇪 The length of a unit vector is determined using the Pythagorean theorem.
• 👷 To construct a unit vector, divide a given vector by its length.
• 🤠 Unit vectors are often represented with a hat symbol to indicate their normalized length.
• 👾 Unit vectors are commonly used in transformations and projections in vector spaces.
• 🎃 In R3, the vectors i, j, and k are unit vectors and serve as the basis vectors.
• 🇦🇪 The standard basis vectors in R3 are equivalent to the unit vectors e1, e2, and e3.

Questions & Answers

Q: What is a unit vector?

A unit vector is a vector with a length of 1, and it is used to represent direction in vector spaces. It is calculated by dividing a given vector by its length.

Q: How is the length of a unit vector determined?

The length of a unit vector is always equal to 1, regardless of the dimension space it belongs to. The length is calculated using the Pythagorean theorem.

Q: How do you construct a unit vector?

To construct a unit vector that goes in the same direction as a given vector, divide the vector by its length. This will normalize the vector and make its length equal to 1.

Q: What is the significance of the hat symbol on a unit vector?

The hat symbol on a vector indicates that it is a unit vector. It distinguishes it from other vectors and indicates that its length is 1.

Summary & Key Takeaways

• A unit vector is a vector with a length of 1 in any dimension space, and its length is calculated using the Pythagorean theorem.

• To construct a unit vector that goes in the same direction as a given vector, divide the vector by its length.

• The length of a unit vector is always equal to 1.

• Unit vectors are often represented with a hat symbol on top of the vector.