This video is about labeling and adding vectors. Vectors are quantities that have both magnitude
(which is just another word for size) and direction. To show their magnitude and direction
we draw vectors as arrows, because arrows also can have a size and a direction. To draw a vector just
draw an arrow pointing in the vector's direction and then write the magnitude of the vector above
it. So as an example if I want to draw a vector I need to give it a direction and a magnitude,
so I'm going to give that arrow a certain size, and then label its magnitude above the arrow like
this. So this vector literally means 10 meters to the right. These are a few more examples
of vectors with specific labels. The arrow's length depends on the magnitude. The bigger the
magnitude, the bigger the arrow, and if you double the size of the magnitude you double the size of
the vector as well. So if this is 10 meters and we want to draw a 20 meter vector, we need to draw an
arrow that's exactly twice as big as that original arrow. The reason why we've proportioned them this
way is that this helps us see the relationship between multiple vectors. Just as an example,
if you're trying to describe motion like distance or displacement using a vector, having the sizes
of the arrows representing the actual sizes of the arrows can help you visualize motion much more
easily than if they were just arbitrary sizes like that so you get a clearer idea of what's going on.
The symbol we use for labeling vectors is a letter with an arrow above it, so that's a variable
that's a stand-in for a vector. So this is vector A. Vector A is equal to 10 meters to the right.
And it's totally arbitrary which letter we use for which vector. As long as we're using a letter
with an arrow above it, that's a variable that stands in for a vector. The same letter without
an arrow above it means the scalar magnitude of that vector. So if I just drew E without an arrow
above it and E with an arrow above it means 10 meters to the right, E by itself just means 10
meters. So that's the notation that we use. So I can see these are all vector quantities and
if I wanted to represent them as scalars, this is how I would do that. Pretty straightforward.
If you multiply a positive scalar by a vector, multiply the scalar by the magnitude of the
vector and keep the direction the same. So as an example, if I multiply this by two, that means I'm
multiplying the magnitude by two but I'm keeping the direction the same ,so two times vector A is
equal to 20 meters because it's two times the magnitude and the direction is the same, but
the vector is now twice as long. If I multiplied the vector by one half, this is what it would look
like. Pretty straightforward. If you multiply a negative scalar by a vector, multiply the scalar
by the magnitude of the vector and flip the vector to face in the opposite direction. So multiplying
by a negative number flips the direction of the vector to aim in the opposite direction. So -2
times A would be 20 meters to the left here. Now we're in the part of the video about adding
vectors. There are rules that make adding vectors different than adding regular numbers, because
vectors also include direction, so the directions of the vectors you're adding are going to matter
for their sum. So the first step to adding vectors is to connect the tip of one vector to the tail of
the other. The tip of the vector is the point of the vector right here. The very front of the arrow
and the tail of the vector is the opposite end of the vector right here. So just remember the point
is the tip and the opposite end is the tail. So we're going to connect the tip of one vector to
the tail of the other, and it does not matter which order we do this in. I'll talk about that
more later, but what matters is that the tip of one is connected to the tail of the other. So if
you watch up here you can see these two ways of connecting the tip and tail are correct,
and these two are incorrect. The connections on the right are incorrect and not how you use the
tip to tail method. So step number 2 is to draw a vector that begins with a tail at the tail of the
first vector and ends with the tip at the tip of the second vector, or the last vector that you're
adding. So if I draw that for these two vectors, the 4 and the 3, I get a new vector that looks
like this. So it starts at the tail of the first and ends at the tip of the last like that. So an
incorrect way of doing this would be to draw it like this. For some reason a lot of students do
this their first time adding vectors, so please make sure that you're drawing from the tail of
the first of the tip of the last. This one's going in the wrong direction. Step number three is to
find the magnitude of the new vector using the magnitudes of the original vectors and geometry.
In this case here specifically, I can see that these vectors are forming a right triangle,
and I remember from math class that if I'm missing the side of a right triangle I can use the
Pythagorean theorem: A^2 + B^2 = C^2, so plugging in my numbers here gets me a final vector value
of 5. We're going to be using the Pythagorean theorem a lot in this class because vectors that
form right triangles come up a lot. We'll learn how to deal with more complicated shapes later.
The main thing I want you to know for now is that you'll have to use geometry to figure out the
length of the missing vector in addition to the other 2 steps. So this is it, this is the answer.
What this is saying is that the two vectors added together, their sum, is equal to this new vector
of 5 pointing in that very specific direction. The direction of the sum matters a lot, so when
you're adding two vectors together you have to consider both their size and their direction,
and use this 3-step process to find what that final vector looks like. Just to give
you an intuition of why these vectors look the way that they do ,why they add together in this way,
we can imagine that you're running along a soccer field. And let's imagine to start off that you run
at a certain speed along the field from one end of the field to the other, and then you run at a new
speed across the field from one side of the field to the other like this. If someone were to tell
you to run in both of those directions at the same time, to perform both of those motions at once,
that would be like adding those two vectors together, adding those two motions together that
you were performing. And if you add both of those vectors together, this is what it looks like. And
so you can see that by adding those two vectors together, we get this new resultant vector. The
blue vector that's not just the numerical sum of the two, it's not just the regular old sum of the
two, we have to use this tip to tail method to find that final value. So that's why we say that
the sum is what it is for vectors. If we have two vectors pointing in the same direction it's very
easy we just do the tip to tail method again: connect tip to tail, draw that new connecting
vector from the beginning of the first to the end of the last, and using geometry. It's pretty
obvious that this has a length of 11. If we have vectors pointing in opposite directions,
again we connect the tip of one to the tail of the other like this so that 2 is kind of pushing back
into the 6 and when I draw that new final vector from the tail of the first to the tip of the last
I can see that this is what it looks like, and so it's basically going to have a length of 6 minus
2 which is equal to 4. So that's how you'd find the size of vectors that are pointing along the
same line. The order that we add regular numbers doesn't matter, and mathematicians have a word for
this. They call this the commutative property, so no matter which order we add multiple numbers it's
always going to come out to be the same resultant number. In the same way vectors actually also
have the commutative property: the order that we add vectors does not matter. Any order we'll get
the same result, so if I add these tip-to-tail it doesn't matter. If I do it like this or like this,
both ways (as long as they are tip-to-tail) yield the same final vector. We're gonna do a quick
check for understanding. You might want to pause the video here and just see if you can determine
the size of each vector sum. I'm going to come back in just a minute and give you the answer.
Okay let's go through connecting these first two tip-to-tail. I can see that 12 and 20 are
just working together so that overall vector is going to be 32 to the right.
Connecting the second 2 tip to tail, this time they're working against each other,
and so that new final connecting vector is going to be 4 pointing up. This next set here I'm going
to connect tip to tail again, and I can see that when I draw that final vector from the tail of
the first to the tip of the last it's forming a right triangle. So if I use the Pythagorean
theorem I find this is equal to 22.36 pointing down like that. Finally, I have these two down
here and so if I connect them tip-to-tail I find that it looks like this, 7.07, after I
use the Pythagorean theorem and that's what you have to know about adding and labeling vectors.