### Learn math, science, English SAT & ACT from

high-quaility study
videos by expert teachers

##### Thank you for watching the preview.

To unlock all 5,300 videos, start your free trial.

# Unit Vectors - Problem 2

###### Norm Prokup

###### Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

One problem you’ll probably see in your homework at some point is find the unit vector with the same direction as v which is equals to <-4, 3> or some vector like that. Unit vector, the definition of a unit vector is that a unit vector has a length of 1. Any vector that has length 1 is a unit vector, but what’s the length of v? It’s the square root of -4² plus 3², which is 16 plus 9 which is 25. Root 25 is 5. The length of v is 5.

In order to get a unit vector, I need a vector that points in the same direction as v, but has 1/5 of the length. My unit vector, I’ll call it u-hat is going to be 1/5 of v. 1/5 of <-4, 3>. That’s <-4/5, 3/5>. This suggests a general method for coming up with the unit vector. For any vector v, a unit vector u with the same direction is, u that is 1 over the magnitude of v times the vector v. So this is a scalar multiple that gives you a length of 1 but the same direction as vector v. Let’s try that out on some problems.

I want to find the unit vector with the same direction as all of these vectors starting with <3, 3>. What’s the length of this vector? It’s the square root of 3² plus –3². So 9 plus 9, root 18 which is 3 root 2. That’s the length. And then the unit vector is going to be 1 over 3 root 2 times this vector. You pull this scalar inside and you get 3 over 3 root 2, -3 over 3 root 2. If you simplify this, you’ll get root 2 over 2, negative root 2 over 2. That’s the unit vector with the same direction as <3, -3>.

What about this guy? V is <-1, -2>. We first find the length. The length of v is the square root of -1² or 1 plus -2², 4. So this is root 5. The unit vector, u-hat, will be 1 root 5, 1 over the length times this vector. That’s going to be <-1 over root 5 and then -2 over root 5>. Your teacher will probably want you to rationalize the denominator. You’ll get negative root 5 over 5 and -2 root 5 over 5.

Finally, let’s try this one. V is <-5, 12>, what’s the magnitude? What’s the length? It’s the square root of -5², 25, plus 12², 144. 25 and 144 are 169 and root 169 is 13. The unit vector is going to be u-hat equals 1 over 13 times <-5, 12>. I distribute the 1/13 over the two components and I get <-5/ 13, 12/13>. That’s it.

It’s really easy to find the unit vector in the direction of any given vector. You first have to take the length or magnitude of the vector, and then you multiply by a scalar, which is the reciprocal of that length, and you get your unit vector.

Please enter your name.

Are you sure you want to delete this comment?

###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

##### Concept (1)

##### Sample Problems (3)

Need help with a problem?

Watch expert teachers solve similar problems.

## Comments (0)

Please Sign in or Sign up to add your comment.

## ·

Delete