calculus 3: vectors, 001
7-week sprint of calc 3, these are just notes
context:
in community college, taking math at ~3x the speed that’s “normal” at other colleges/unis. (yes, the classes are for a letter grade).
did calc 1 & calc 2 in 8 weeks. each. also comp sci 101 (microsoft, html, & python) during the time.
calc 3 is technically a 5–week sprint.
but i got two weeks before the class starts.
so imma use this time to get ahead of the game.
and i know we won’t cover all of calc 3 in that class, just the most important pieces of it.
personally, in these two weeks…
i’m only looking at concepts of calc III that would be useful for ML, DL, linear algebra, etc.
ML → machine learning
DL → deep learning
ok, cool.
expect an 2 articles per week.
p.s. — yes, linear alg. > calc 3 in terms of usefulness for ml/dl, but imma follow the sequence, tho i’ll teach myself some linear, why not? won’t kill me.
p.p.s. - thx prof. leonard. icon u are.
11.1 vectors
vectors are this line in space u can draw.
normal lines (from calc 1 & calc 2) only have a direction.
vectors have direction AND a speed.
speed, in terms of calc 3… is known as magnitude.
magnitude is synonymous with the word “length” in calc 3.
so, in layman’s terms: vectors have a direction and a specific length (speed).
the notation: the letter ‘v’ with an arrow on top of it.
and if u want to describe going from one point to another:
ex: from point A to point B… u do this: AB, with an arrow.
which letter comes first?
^from the letter u start at.
but vectors can have different directions & different length.
but some look so similar to another one, it just either:
- goes in a different/opposite direction of the vector you’re considering
- is shorter/longer than the vector you’re considering
btw, if a vector is negative,
it means it’s going in the opposite direction of the original vector.
mini-note:
if it helps, pretend vector “v” was equal to 1.
substitue “v” for 1, so u can see why each vector is bigger/smaller or changes direction from the original vector “v”.
all these vectors are the same one as vector “v” btw. (aka the first vector on the image, far left).
just some are bigger, smaller, or change direction.
but notice how they’re all parallel?
because of this, they’re given a specific name: scalars.
all scalar vectors will alwys be parallel.
(there’s a theorem for that, i’ll show either in this article or the next one).
properties of vectors, tl;dr
- you can add vectors
- and you can subtract vectors
let’s say we have two vectors: one is “a” & other one is “b”.
both are straight-line vectors.
“a” goes up. “b” goes to the right.
if u add both vectors: it’s this: “a” + “b”
so u attach the ends of both endpoints at the endpoint of vector “a”.
it’s like playing “connecting the dots” lol
but if it’s “a” - “b”:
picture it as this … “a” + “-b”
those two equations are the same thing.
but the negative vector, means that the original vector “b”, going to the right, must now go in the opposite direction… to the left.
so instead of your “b” vector going right, it will go left.
special cases of vectors
yes, there are unique ones that have their own name and notation.
some vectors start at the origin. (0,0)
these vectors are known as position vectors.
they got these brackets to note them:
<V1, V2>
the brackets “<>” mean this: the vector starts at the origin (0,0).
but sometimes, u see this: (V1,V2)
they are NOT the same.
this “()” means that it’s a point on the position vector (line) “<>”.
in terms of formulas to find length, slope, etc., i’ll make recordings of some problems i do, instead of writing, prefer the video.
i’ll just talk about the basics and terms for now, ok thx.
unit vector: a vector with a length/magnitude of “1”.
unit vectors are critical.
they tell us in what direction the vector is going.
irl, it’s the equivalent of saying: i drove 60 miles north from sf.
11.2 vectors in 3-d (notes only, no equations/formulas)
i’ll upload the equations and proofs in vids instead.
vectors in 3-d only have one difference:
you add a third axis: the z-axis.
if u having trouble picturing this in 3-d:
picture this:
the z-axis is going through the device u’re viewing this text.
the x & y axis don’t change from regular calc problems.
but if u were to walk along the z-axis, it’s like if u were walking diagonally.
on the x-axis u walk right or left.
y-axis, it’s up & down.
here’a dope vid on it, if u need to visualize it:
ok, the rest of the notes will be turned into videos.
i’ll add them into the article in t-minus 24 hrs.
CJ
vids
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