3D Curves and their Tangents | Intro to Vector-Valued Functions

A space curve, or vector-valued function, is a function with a single input t and multiple outputs x(t), y(t), z(t). In this video we introduce these functions and see how to visualize them in 3D. We then turn to the problem of tangent lines to these types of curves. The definition is modeled after tangents to graphs of functions.
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[@tramquangpho]

The hardest part in teaching calculus is making everything seem obvious, “ but many people fail to do that. Great teaching indeed . One question : you say , the vector r(t+ delta(t) - r(t) is increase in size when divide by t less than 1; but the more t decease and that vector really close to tangent vector , I dont see it increase in size anymore ?

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[@lehou8320]

Dr. Trefor, thank you very much for the lessons. They're really amazing! So... dr/dt is the velocity (vector v), not the tangent vector T, right? dr/ds is T.

[@calcubite9298]

at 5:47 is the yellow vector pointing in the wrong direction? I was taught that when are doing (vector A) minus vector B, when A and B have a common starting point, in this case, the origin, that the (vector a minus b) starts at the tip of B and points TOWARDS vector A. Here in this picture, the vector starts at the tip of A and points towards vector B.

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[@212ntruesdale]

In single variable, the derivative is the slope of the tangent line. When it comes to vector valued functions, is the magnitude of the tangent vector analogous to the slope of the tangent line? So, in the case of a straight line, the tangent vector would just be the zero vector?

[@DrTrefor]

Exactly. Another way to think of it is that it is the "speed" aka the length of the velocity vector

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[@pa7832]

Dr. Bazett, what 3D visualization tool do you use? It seems useful as you are able to change your viewing angle etc.

[@DrTrefor]

Everything here is done in MATLAB

[@pa7832]

Thanks Dr. Bazett! Your videos with the animations are super useful!

[@TheBgracey]

In the formula - r (t) = cos(t)I + sin(t)j + tK Why is a plus sign used? Is the output to be thought of as a vector or as a scalar?

[@jgkostas]

Dr T -- Can you recommend a multi-variable calculus text? I prefer texts will physical explanations and examples. Thanks for the spectacular vectors in 3-space in the videos....

[@dylanico887]

At 2:30, why do you just randomly consider x^2 + y^2 = 1? Aren't the x and y components respectively cost and sent?

[@wakeawake2950]

Thank you sir for this video, I want to ask is this the gradient of (r) vector?

[@DrTrefor]

Gradient only applies to a scalar function

[@wakeawake2950]

@@DrTrefor thnk u sir🙏

[@andrewwoan]

Hi! what software do you use to graph in 3 dimensions like that?

[@DrTrefor]

I use MATLAB for most things, but a good option for students to play with that is free is geogebra

[@andrewwoan]

@@DrTrefor Thank you very much Dr. Bazett!

[@franekkos574]

🙏🏼

[@thecarlostheory]

May you please give a geometric or formal derivation of delta t, instead of putting it as a scalar that increase the delta r magnitude vector? For me that feels very artificial, and I don´t like it... Is there another way to derive delta t participation in the equation of tangent vector?

[@Caleepo]

hello Trefor Bazett, thank you for your nice video, but Ive got a question somehow. How can you exactly know if two different inputs have got the same output. How can you be sure you are dealing with a function ?? If two inputs have got the same output then how can you see it/ calculate it ?

[@Caleepo]

@@DrTrefor ah thank you very much Im on my first year of engineering it really made a mess of my head, can't think properly as I used to do in highschool lol

[@inquisitiverakib5844]

Tangent Vector 3:27

[@marcbanfi4985]

Can a vector-valued function be in only two dimension?

[@rajisingh5710]

Sir what will be unit tangent vector

[@samisiddiqi5411]

9:14 I think we should be taught the Leibniz Notation more in school...

[@nomann5244]

what is the name of the software used to plot the 3d graph

[@DrTrefor]

I use MATLAB, but lots of options that work great too

[@nomann5244]

Thank you for your reply and answer.

[@212ntruesdale]

The fact that the yellow vector stays yellow even after being multiplied by a scalar confused me. The vector that becomes the tangent vector and delta r are not the same. (You do erase the expression for delta r, but I didn’t get as much from that as I would have from a change in colors, to emphasize the fact that the two vectors aren’t the same.) I don’t think this is quibbling either, when the magnitude of the tangent vector has such an amazing interpretation, that being speed when t is time.

[@anishmanandhar1203]

Nicely explained. Would be better if you could explain via delta-epsilon theorem too

[@satyampanchal-1016]

how do you plot these vector valued functions on 3D?, i mean what software u use?

[@DrTrefor]

I use MATLAB

[@satyampanchal-1016]

@@DrTrefor i'm a physics student in first year of my under graduation, and i'm looking for such resources, can you briefly tell me what should I do on Matlab? i don't know if its very advanced for someone like me do I need to learn how to use it (like some computer language), actually i have zero clue about this Matlab , can you nudge me in the right direction?

[@geoffreyfaust3443]

"Leibniz" (as in, derivative notation) should be pronounced like the long i in 'like'

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Why a vector valued function should have only 1 input?

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Where is the problem solving????

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[@DrTrefor]

ha, I spent three years in the US and now I'm mentally stuck using both:D

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