Constructing a unit normal vector to a curve | Multivariable Calculus | Khan Academy

Figuring out a unit normal vector at any point along a curve defined by a position vector function

Missed the previous lesson?

Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

For free. For everyone. Forever. #YouCanLearnAnything

Subscribe to KhanAcademy’s Multivariable Calculus channel:

[vladislavmatiusenco]

If anyone is wonderind the formal explanation of why the nomal vector is n = (dy/dt)î - (dx/dt)ĵ the explanation is the following:
the explanation has to do with rotation, therefore with sine and cosine functions. If we express dy/dt and dx/dt in terms of the angle of rotation and of the radius, that means in polar coordinates


(x, y)->(rcosθ, rsinθ),


then we have


dx/dt = rcosθ
dy/dt = rsinθ.


The normal vector is the tangent vector rotated 90° clockwise, that rotation corresponds to -π/2 rad = 3π/2 rad. We know from these trigonometric relations that


cos(3π/2+θ) = sinθ
sin(3π/2+θ) = -cosθ.


So the coordinates of the normal verctor, which is the tangent vector rotated 90° clockwise are


dx/dt = rcos(3π/2+θ) = rsinθ = dy/dt
dy/dt = rsin(3π/2+θ) = -rcosθ = -dx/dt


So here we have the conclusion n = rsinθî - rcosθĵ = (dy/dt)î - (dx/dt)ĵ

[alejrandom]

Another way to look at this is just applying a rotation matrix
0 1
-1 0
To the normalized derivative of r
<x'(t), y'(t)>
The result is <y'(t), -x'(t)> scaled by 1/||v(t)|| (where v is the tangential velocity vector, or dr/dt).

[Yakushii]

Sooo many physics topics is largely based around vector calculus. After all, tons of physics measurements are indeed vector values. Force, magnetic field, electric fields, displacement, acceleration, momentum, angular momentum, velocity... So yeah, this is totally physics, it's just the "mathy" part of physics ;)

[purefrlyfe]

been waiting for the rest of the vector calc for ages, I LOVE YOU! :)

[ossigaming]

I got this perfectly with my 16 years, which kinda makes me sad, since other learned this already at age 15. I'm way too late already. And the sad thing is, I got to do it in 1 week. Not bragging, but. . . trying to act like a nerd. Anyway, s much thanks to your course. Apart from the problem with the cursor once, absolutely everything is alright!!!

[ThexScreamingxHead]

Is there a sign error in the final answer? Shouldn't the unit normal point pi/2 rads CCW perpendicular to the unit speed curve?

[cappy]

Isn’t it also possible to just convert the vector from the origin into a unit vector, and then multiply it by the magnitude of the tangent vector?

[amansingh]

How do we decide the orientation of normal vector

[karimkhan]

wonderful-- can u teach special functions of physics or tensors or group theory for physics

[gnaples]

Mr. kahn - Whats the best text to use for a businessman to use for business mathematics in your opiion?

[scredB]

I love your vids! they are sooo educational; please keep it up! xx

[Tincek]

Shouldn't n be dx*j - dy*i the way you drew it?

[fabriziamugnatto]

i still don't get this. but i'm probably not old enough yet. whatgrade do you lewrn this in cause i

[ThexScreamingxHead]

Nvm. I just noticed that you wanted to focus on the other one...