Angle between vectors leads to defining the Dot Product | Multivariable Calculus

Given two vectors, what is the angle between them? By applying the cosine law we are led to a formula. From this we define the dot product. At one level, the dot product is just a component of that angle formula. But it is a component that is used often enough, and important enough in a range of applications, that we give it it's own name: the Dot Product. This is the first video in my brand new playlist: Calculus III.

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This video was created by Dr. Trefor Bazett

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[DiegoMathemagician]

u_3 and v_3 must be squared, not cubed. Anyway, great video!

[davidf]

FINALLY someone focusing equally on the motivation as on the mechanics ! Thanks very much Dr. Bazett !

[blootybloot]

You just made clear in one minute what a hundred regurgitations of the dot product proof couldn't do for me; the way to separated the squares from the spliced vector components made it so visual and intuitive!

[まつまつ-xf]

There is a concept called "Duality" in the dot product. I think this is really amazing and profound to discover the natural meaning behind math. Also, the orthogonality is significant as well in Linear algebra. I mean, everything seems to get connected just with the central and easy math notation!! Humans are incredible!!!

I am not good at math, but I found myself starting to be fond of math. It really feels like magic!!!

[scientificresearch]

Really I am search for From long a time now I found you 🙏

[Jack_Callcott_AU]

This video is very helpful. I have done a course in multivariable calculus a long time ago and I'm very rusty, but this explanation of a derivation of the dot product is very enlightening. Thank you Doc. Bazett.

[vivekracing]

Many textbooks directly define the dot product and proceed from there. You, sir, derived it from the cosine rule. Thank you for this insight.

[alkankondo]

Great video! Unique idea to the Law of Cosines to introduce the concept of the dot product!

[perseusgeorgiadis]

Tried to watch that before learning linear algebra. Now that I have the basics down, it’s so much more satisfying

[NuclearPhysix]

The best explanation of the dot product I've seen

[ntruesdale]

So the ‘dot product’ is just a test to determine whether two vectors in space are orthogonal. The cool thing is that it’s a super simple test. Pretty hard to look at two vectors in 3D and say whether they are orthogonal, but everybody knows how to multiply and add! It needs to be said that the math leading to the dot product is not particularly easy. But the dot product is all you need to know to perform the test.

[raom.okasha]

finally my long awaited qs is answered, finaly I know WHY we do dot product the way we do it, it is surprisingly simple but no teacher taught this, instead i just knew how to perform it thank you

[ElR]

I have so far hated Calc 3, largely because I'm not actually sure what I'm doing or why I'm doing it. I mean, I have an A heading towards our final exam and I can regurgitate like the best of them, but I hate the class because the material feels vague and messy. Thank you for making a series that explains what and why!

[GEMINDIGO]

Thanks so much for taking the time to show us this Trefor.I look forward to the other videos

[gusthonnings]

Trefor, coming all the way from Canada to help me out with mulitvariable calc. Wish you were still at UC!

[brianarmor]

You're a very good teacher. You deserve better audio quality!

[dantecata]

Trefor you are saving my life with these videos, thank you! We miss you at UC!

[anli]

Truly great. I've been watching all of your content. Thank you! Also, great tan in this video!

[rockyjoe]

OMG MULTIVARIABLE CALCULUS AMAZING KEEP IT

[luyombojonathan]

Interesting stuff !!! Not yet able to understand but I will keep on rewatching