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### Course: Linear algebra>Unit 2

Lesson 2: Linear transformation examples

# Unit vectors

What unit vectors are and how to construct them. Created by Sal Khan.

## Want to join the conversation?

• In this video, Sal indicates that another name for a unit vector is a normalized vector.
However, a normal vector does not necessarily have a length of one, and it does not go in the same direction as the original vector, rather it is perpendicular to it.
Is anyone else confused by this terminology?
• It is unfortunate that the words look similar, but "normalized" and "normal" are two very different things.

"Normalized" is used as a verb and means, "force a vector to have length 1." This is easily accomplished by dividing the vector by its magnitude.

For example, the magnitude of 3i + 4j is 5, so the normalized version of this vector is (3/5)i + (4/5)j. The normalized version of a vector always has length 1 and is parallel to the original vector.

On the other hand, as you point out, a normal vector is something completely different, and is always perpendicular to the vector it is normal to!

Ugh, not the best of terminology, but that is life in linear algebra.
• are there any other notations for vectors?
• i j k notation. See the physics playlist.
• What's the point of a unit vector, why do we need these at all?
• Sal normally gives a proof that what he's saying is true, but he doesn't here. It doesn't just "pop out at me", as he would say, that û = 1/||v|| v for any given vector v. Is there some way to prove this?
• Think of it like this. All vectors can be represented as a unit vector times a scalar quantity (to up scale the vector). Therefore we can write u-hat*||v||=v. Dividing both sides by ||v|| gives u-hat=v/||v|| which is the same as u-hat=1/||v|| v
• In vectors , what we mean by saying " value of vector " ?
And what is the difference between the value of vector and the magnitude of vector ??
Thank you .
• The magnitude of the vector is equivalent to the length described in this video. I'm not sure that I know of anything being the "value of vector". There are Vector-valued functions, which is just a function that returns vectors, but that is a different concept than what is in this video.
• does a zero vector has a magnitude ??
• The magnitude of a zero vector is, as the name implies, 0.
• i have no clue whats going on... HELP!
• A unit vector is any vector where || v || = 1. Another way to say it is that a unit vector is exactly one unit long. Thus, "unit vector" (one unit). If that's not what you needed, please specify a bit more what you're confused about.
• I have no idea what a vector is.I never heard of one before.But i am going to be a scientist and scientists learn different things!! :(
• That's the spirit!

A vector is a quantity that has both a magnitude and a direction. They are used to represent a lot of things that single numbers (called scalars) cannot, and are incredible useful not only in math, but also in physics, computing, economy, etc.

Vectors are used extensibly in Linear Algebra, but I suggest you familiarise with them before continuing with Linear Algebra, since it's assume that you have some knowledge of vectors. Here are the introductory videos for vectors: https://www.khanacademy.org/math/precalculus/vectors-precalc/vector-basic/v/vector-representations-example