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## Linear algebra

### Unit 1: Lesson 1

Vectors- Vector intro for linear algebra
- Real coordinate spaces
- Adding vectors algebraically & graphically
- Multiplying a vector by a scalar
- Vector examples
- Scalar multiplication
- Unit vectors intro
- Unit vectors
- Add vectors
- Add vectors: magnitude & direction to component
- Parametric representations of lines

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# Adding vectors algebraically & graphically

To add the vectors (x₁,y₁) and (x₂,y₂), we add the corresponding components from each vector: (x₁+x₂,y₁+y₂). Here's a concrete example: the sum of (2,4) and (1,5) is (2+1,4+5), which is (3,9). There's also a nice graphical way to add vectors, and the two ways will always result in the same vector.. Created by Sal Khan.

## Want to join the conversation?

- Can we subtract vectors?(91 votes)
- Yes you can and it works in exactly the same way. That is subtracting the vector [3, -4] would be the same as adding the vector [-3, 4]. So subtracting a vector is the same as adding a vector that goes in the opposite direction with the same magnitude. It's a bit like when you first learn subtraction using a number line and see that subtracting a number mean moving left along the line whereas adding means moving right.(190 votes)

- So why the magnitude of a vector doesn't imply its origin; you say that we don't care about where the vector starts, but the magnitude, in order to be defined, doesn't need to have mentioned a point of departure and a point of arrival?(18 votes)
- The starting and ending points of a vector don't matter as long as its length and direction are constant. For example, you could have a vector of length 5 in the vertical direction starting from the origin, or any other point.(10 votes)

- At1:06, what is meant by the sentence that vector a and vector b belong to R2. What is meant by R2? Please i want an immediate response!!(9 votes)
- R2 represents all the real numbers in a 2-Dimensional world.

So any RN represents all the real numbers in a N-Dimensional world.

Hope this helps,

- Convenient Colleague

Sorry that I get here 5 years late...(26 votes)

- What is rectangular form!?(16 votes)
- A vector in rectangular form is when you specify the components of the vector along each of the rectangular cartesian coordinate system axis, so you can specify a vector as a tuple of numbers:
`(a, b)`

, or using unit vectors along the axis:`ai + bj`

.

Another way of representing vectors is in polar notation, where you give the length of the vector and it's angle measured form the positive x axis, that is usually called "polar form".(21 votes)

- At6:43and before Sal show different vectors and how you can draw them. Is this really necessary to know how to draw all these different vectors? on the "graph"or can I just draw the vector i am most comfortable with.(12 votes)
- He is showing how you can draw the same vector on different coordinates (better to say the same vector scaled for a given scalar). It is not necessary to draw all those different vectors, but it is useful to know you can draw them that way. Also, when you solve problems which include the addition, substitution etc. of vectors you are always going to draw the vectors which have certain coordinates or given dots (A, B, C...). You can always "put" them on the position where you need them to be.(9 votes)

- Does the vector always start at the origin?(9 votes)
- It doesn't have to start at the origin because it doesn't matter where it starts. A vector doesn't have a "starting point" or "ending point". It only has a magnitude and direction. A vector that is REPRESENTED as starting at the origin is the same vector that is REPRESENTED as starting anywhere else, as long as the magnitude and direction are the same. Note that vectors aren't really equivalent vectors, but the same vectors. They are just represented at different places.(2 votes)

- Here we form a triangle by using head to tail rule. When do we use the parallelogram rule?(6 votes)
- The area of the parallelogram formed by 2 vectors is the magnitude of the cross product of those vectors. Otherwise, the parallelogram can visually represent the commutative property of vector addition (a+b = b+a).(8 votes)

- Exactly how can you use vectors in the real world?(5 votes)
- Well, vectors are basically everywhere! When we kick a ball in a projectile motion, the velocity of the ball is a vector (it rhymes). Programmers also use vectors in their programs so that the animation of objects look real. NASA uses 3D vectors to plot courses for their space exploration robots.(7 votes)

- how do you find the magnitude of a vector?(4 votes)
- You can think of it as finding the hypotenuse in a right triangle. For example, we can have a vector "v" that begins at the origin and terminates at point (-5, 12). We can create a right triangle in which the vector is the hypotenuse, so we can use the Pythagorean Theorem.

c^2 = a^2 + b^2

c = sqrt(a^2 + b^2)

|v| = sqrt(x^2 + y^2)

|v| = sqrt((-5)^2 + 12^2)

|v| = sqrt(25 + 144)

|v| = sqrt(169)

|v| = 13(3 votes)

- At about1:30Sal writes a bunch of stuff I don't understand. What is the meaning?I was also wondering why are there braces around the vectors x and y.(4 votes)
- These are 2 vectors in column (vertical) form, and their sum.(1 vote)

## Video transcript

So I have two
2-dimensional vectors right over here,
vector a and vector b. And what I want to think
about is how can we define or what would be
a reasonable way to define the sum of
vector a plus vector b? Well, one thing that
might jump at your mind is, look, well, each of
these are two dimensional. They both have two components. Why don't we just add the
corresponding components? So for the sum, why don't
we make the first component of the sum just a sum of
the first two components of these two vectors. So why don't we just make
it 6 plus negative 4? Well, 6 plus negative
4 is equal to 2. And why don't we just
make the second component the sum of the two
second components? So negative 2 plus 4
is also equal to 2. So we start with two
2-dimensional vectors. You add them together,
you get another two 2-dimensional vectors. If you think about it in terms
of real coordinate spaces, both of these are
members of R2-- I'll write this down here just
so we get used to the notation. So vector a and vector b
are both members of R2, which is just
another way of saying that these are both two tuples. They are both two-dimensional
vectors right over here. Now, this might make sense just
looking at how we represented it, but how does
this actually make visual or conceptual sense? And to do that, let's
actually plot these vectors. Let's try to represent
these vectors in some way. Let's try to visualize them. So vector a, we could
visualize, this tells us how far this vector
moves in each of these directions--
horizontal direction and vertical direction. So if we put the,
I guess you could say the tail of the vector
at the origin-- remember, we don't have to put
the tail at the origin, but that might make it a little
bit easier for us to draw it. We'll go 6 in the
horizontal direction. 1, 2, 3, 4, 5, 6. And then negative
2 in the vertical. So negative 2. So vector a could
look like this. Vector a looks like that. And once again,
the important thing is the magnitude
and the direction. The magnitude is represented
by the length of this vector. And the direction
is the direction that it is pointed in. And also just to emphasize,
I could have drawn vector a like that or I could
have put it over here. These are all
equivalent vectors. These are all equal to vector a. All I really care about is the
magnitude and the direction. So with that in mind,
let's also draw vector b. Vector b in the horizontal
direction goes negative 4-- 1, 2, 3, 4, and in the vertical
direction goes 4-- 1, 2, 3, 4. So its tail if we
start at the origin, if its tail is at
the origin, its head would be at negative 4, 4. So let me draw that
just like that. So that right over
here is vector b. And once again, vector b
we could draw it like that or we could draw it-- let me
copy and let me paste it-- so this would also be
another way to draw vector b. Once again, what I
really care about is its magnitude
and its direction. All of these green vectors
have the same magnitude. They all have the same
length and they all have the same direction. So how does the way that
I drew vector a and b gel with what its sum is? So let me draw
its sum like this. Let me draw its sum
in this blue color. So the sum based on this
definition we just used, the vector addition
would be 2, 2. So 2, 2. So it would look
something like this. So how does this make
sense that the sum, that this purple vector
plus this green vector is somehow going to be
equal to this blue vector? I encourage you
to pause the video and think about if
that even makes sense. Well, one way to think about
it is this first purple vector, it shifts us this much. It takes us from this
point to that point. And so if we were to add it,
let's start at this point and put the green
vector's tail right there and see where it
ends up putting us. So the green vector, we
already have a version. So once again, we
start the origin. Vector a takes us there. Now, let's start over
there with the green vector and see where green
vector takes us. And this makes sense. Vector a plus vector b. Put the tail of vector b
at the head of vector a. So if you were to
start at the origin, vector a takes you
there then if you add on what vector b takes you,
it takes you right over there. So relative to the
origin, how much did you-- I guess you could say-- shift? And once again,
vectors don't only apply to things
like displacement. It can apply to velocity. It can apply to
actual acceleration. It can apply to a
whole series of things, but when you
visualize it this way, you see that it does
make complete sense. This blue vector,
the sum of the two, is what results where
you start with vector a. At that point right over there,
vector a takes you there, then you take vector b's
tail, start over there and it takes you to
the tip of the sum. Now, one question you
might be having is well, vector a plus vector
b is this, but what is vector b plus vector a? Does this still work? Well, based on the
definition we had where you add the
corresponding components, you're still going to
get the same sum vector. So it should come out the same. So this will just be
negative 4 plus 6 is 2. 4 plus negative 2 is 2. But does that make visual sense? So if we start with vector b. So let's say you
start right over here. Vector b takes you
right over there. And then if you were
to go there and you were to start with vector
a-- so let's do that. So actually, let me make this
a little bit-- actually, let me start with a new vector b. So let's say that that's our
vector b right over there. And then-- actually,
let me give this a place where I'll have
some space to work with. So let's say that's my
vector b right over there. And then let me get a
copy of the vector a. That's a good one. So copy and let me paste it. So I could put vector a's
tail at the tip of vector b, and then it'll take
me right over there. So if I start right over
here, vector b takes me there. And now I'm adding to that
vector a, which starting here will take me there. And so from my original starting
position, I have gone this far. Now, what is this vector? Well, this is exactly
the vector 2, 2. Or another way of
thinking about it, this vector shifts you 2 in
the horizontal direction and 2 in the vertical direction. So either way, you're going
to get the same result, and that should, hopefully,
make visual or conceptual sense as well.