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### Course: Multivariable calculus>Unit 5

Lesson 4: 2D divergence theorem

# Constructing a unit normal vector to a curve

Figuring out a unit normal vector at any point along a curve defined by a position vector function. Created by Sal Khan.

## Want to join the conversation?

• Calculating the normal unit vector seems pretty straightforward. What is the purpose of finding it? Why kind of applications would you use this in?
• I realize it was two years ago but I'll answer the question anyways if someone is wondering. The purpose of a unit vector is to find the direction in which a vector is traveling in (its magnitude is one.) With this, you can manipulate it and other vectors to have them travel in same direction or different directions easier. A calculus IV concept I recently learned was with a gradient of a function, knowing that a unit vector in the same direction as the gradient would give the maximum possible change of the gradient.
• I am thinking of a proof of the unit normal vector being equaled to that expression using dot product. Actually, I have done it using mspaint lol.
Here is my proof:
http://postimg.org/image/nuj1gp3xb/
http://postimg.org/image/bjnislcmh/
I think it is fairly easy to do it and to understand it once you have learnt vector calculus. (Although It took me an hour of trial and error to figure out this proof, I tried using gradient to prove it but failed. ) May I ask if you want to or not to include this or any similar proofs in your video guides? I guess this will be helpful for the audience.
• Does it really matter which component you make negative? There is a khan academy article on constructing unit normal vectors to curves in the section about vector line integrals. This article makes the opposite component (the i component) negative.
• Isn't a tangent vector dr/dt rather than just dr? Why specifially dr/dt cannot be a tangent vector to a curve?
• at ~, I thought that the normal vector always points in the direction the curve is curving. So wouldnt the unit normal vector point in the opposite direction (towards the origin)?
• The normal vector is defined as any vector which is perpendicular to the curve. Hence the vector you're suggesting which points to the origin would also be described as a normal vector. In this case he is simply taking the outward pointing vector without having disambiguated as one would expect if we were to be strict. You can read more about this on MathWorld: http://mathworld.wolfram.com/NormalVector.html
(1 vote)
• Why is -dxj going up when it should be going down since it is negative?
• Why only -dx j_ ? Why dy_i doesn't get a minus sign, where normal vector = dy i_ - dx _j?
• You actually get both, +dy î - dx ĵ and -dy î + dx ĵ are both normal vector to the curve.
• A couple things: Transforming dxi + dyj into dyi - dxj seems very much like taking a determinant. What's the relation? And two, couldn't you find a unit normal vector by finding the unit tangent vector, then making a vector perpendicular to it? i.e., using dot product to find perpendicular vector, or using a different vector and subtracting its projection onto the previous one?
• Yes you can do the transformation (rotation) using the rotational matrix. We can write dx î + dy ĵ as row vector, and cross it with the rotational matrix. 𝜃=-𝜋/2 if the curve is positively oriented (anti-clockwise), 𝜃=𝜋/2 if the curve is negatively oriented (clockwise).
So for positively oriented curve,
| dx dy | X | cos(-𝜋/2) -sin(-𝜋/2) | = | dx dy | X | 0 1 | = | -dy dx |
________| sin(-𝜋/2) cos(-𝜋/2) |__________| -1 0 |
So we obtain, -dy î + dx ĵ
Similarly, for negatively oriented curve , we obtain dy î - dx ĵ
Check http://en.wikipedia.org/wiki/Rotation_matrix for more details
EDIT: I have used dot product to find perpendicular vector.
Here is my proof:
http://postimg.org/image/nuj1gp3xb/
http://postimg.org/image/bjnislcmh/
• Does anyone know what this symbol means?

ψ

I found it in an equation that ran like so: a ψ b = a^2 - 2b. Any help would be appreciated, as I've asked everyone I know and no one seems to have any idea. Thanks.
(1 vote)
• That sounds like a question that you see on the SAT every now and then. They just define a new operation (along the lines of +, *, ^, etc.) and tell you what it does when you perform the operation with two given numbers. In other words, this symbol used for this purpose is unique to this problem and is not something where you could ask a random math professor at Princeton or something and they would tell you, "Oh yeah, that means a^2 - 2b."

It is the Greek letter psi, though. It makes a sound of "ps," and is the first letter of, for example, the Greek word from which the English word "psychology" is derived.