The meaning of the dot product | Linear algebra makes sense

This video explains why the dot product is about how much vectors point the same way.

[Improbabilities]

Just for the record, I do notice the effort you put into these videos. You're doing great! Keep it up!

[shredda]

literally the thumbnail alone explained it better than everyone else

[KHANPIN]

Honestly, I wasn't expecting those pauses in the video, but turns out, they helped a lot with just grasping the concept and understanding. Thanks!

[FortniteMaster-viqt]

This is incredible, taking a concept that confused me to no end and helping me understand in 13 min. Thank you so much

[lucasfreitag]

Finally a video on YouTube that gives a really intuitive visualisation of the dot product. Well done. Thank you.

[tejashreebhat]

The amount of creativity in this video, the amount of efforts you have put in editing this. Salute. 🔥🔥

[bijannatividad]

I've been looking at a lot of videos and forums looking for the practical meaning of a dot product. Every video and forum I found only defined what dot product is and they were very technical about it. This video gave me the answer after 5 seconds. Thank you!

[sanjaisrao]

I cant even imagine how much work you did for this video, its great

[Anna-qpfi]

The first 8 seconds literally answered a question I've been trying to solve for the last 20 minutes with multiple websites and textbooks.

[no_scope]

this is actually so sick, a lot of work went into this and it really did help me

[JohnWalz]

You're videos are amazing! Really helped me understand super daunting subjects. You're ability to break these topics down and explain them in a straightforward way is so awesome!!!

[acborgia]

For Homework 2:
If we take v_b as the unit vector on its axis,
from the previous exercise we know that
x = v_u . v_b / ||v_b|| and y = v_v . v_b / ||v_b||
Now we add v_u and v_v
v_u + v_v = (x+y) * v_b + perp
We apply the same theorem on x+y as on x and y
(x+y) = (v_u + v_v) . v_b / [[v_b||
Now replace x and y by their expressions (the ||v_b||s cancel out)
v_u . v_b + v_v . v_b = (v_u + v_v) . v_b (1)
This shows the distributivity of the dot product

(v_a * k) . (v_b . s) = ||v_a * k|| * ||v_b*s|| * cos(theta) = k * s * ||v_a|| * ||v_b|| * cos(theta) = k * s * v_a . v_s
This shows the scalar multiplication property of the dot product (not sure if that's the right term lol)

v_u = alpha * v_v1
v_v = beta * v_v2
Now we replace v_v and v_u in the formula (1)

(alpha * v_v1 ) . v_b + (beta * v_v2) . v_b = (alpha * v_v1 + beta * v_v2) . v_b

From last property this is the same as:

alpha * v_v1 . v_b + beta * v_v2 . v_b = (alpha * v_v1 + beta * v_v2) . v_b
This shows the dot product is bilinear

Homework 2 finished

[roger]

Suddenly discovered this channel. Loving it!

[harrytresor]

Answer for the last question:
So we have vectors a and b that can be written as a1x + a2y for vector a and b1x and b2y for b. Given the linearity formula for the dot product. a•b = (a1x + a2y)•(b)
= a1x•b + a2y•b
Substitute for b and we get:
a1x•(b1x + b2y) + a2y•(b1x + b2y)
a1x•b1x + a1x•b2y + a2y•b1x + a2y•b2y
The dot products of orthogonal vectors is 0 so the equation simplifies to:
a1x•b1x + a2y•b2y
Since the are facing in the same directions, the is the same as multiplying their magnitudes together. Thus
a•b = a1b1 + a2b2 ...

[niranjanarunkshirsagar]

Amazing! Our human brain is evolved such a way that we understand anything very easily if it is intuitive & your techniques are so intuitive and gripping that anyone who is very weak in maths can learn by heart the very concept behind "dot product". In that way, your techniques can be called "compatible with evolution". Richard Feynman also had this ability. Go ahead!

[aion]

Wow. Finally i understand whats the point of the dot product. Should be called "similarity product" or something to point to it's meaning. Can't believe i went on all this years not understanding this basic idea.

[usernameisamyth]

Ain't u gonna extend the playlist?
I loved ur explanation.

[pilotomeuepiculiares]

WOW.
This video is better than any classes I had in school or college.
I'll watch again later and do the "youtube homeworks" in paper and not half-way in my mind.
Thanks

[sanjalisubash]

Wow! I just started linear algebra and was already confused as to why dot products matter. The way you explain this is so good and you can really make out the effort in the videos. I can't believe I haven't seen your content before! Subbed right away!!

[fcti]

Outstanding explanation of the dot product! This is the first video-and I've seen plenty- that's helped me get an intuition for this concept. Thanks so much!