Hey guys in this video we're going to be introduced to two of the most important
tools in a physicist's mathematical toolbox: vectors and scalars. So let's
dive right into it. So what is a vector? Well vectors are mathematical quantities
that are fully described by a magnitude and a direction. What do we mean by
magnitude? Well, that's just really the size or quantity of something. If you can have
more or less of something, if it can be measured on a scale, or if it can be
bigger or smaller, then it has a magnitude. And what about direction?
Well, that's simple. It's an indication of how something is oriented in space. Examples
are right and left, up and down, or even east and west, as well as north and south.
In mathematics variables that are vectors usually have an arrow on top of
them. So if the variable V were a vector we'd put an arrow above it like this to
represent that it is in fact a vector quantity. Great! So here is a simple
example of a vector. Suppose a friend is asking how to get to the grocery store
and we tell them it's half a kilometer north of here. At a glance, it might look
like we just gave them one piece of information telling them where to go but
in reality it actually consists of two pieces of information. First, we told them
it's half a kilometer from here. Second, we also mention that it's north of here.
The first piece of information was the distance. It can be bigger or smaller.
It's a quantity, or in other words, a magnitude. The second piece was obviously
the direction and hey remember what a vector is? It's something that has magnitude
and direction. So when we said it's half a kilometer north of here, we were really
giving our friend the vector: zero point five kilometers north. Awesome! All right,
now say another friend comes to visit and they notice that you've got 42 books
on your bookshelf. So the number of books that you have ... is it considered a
vector quantity? Well, the answer is no. You see, even though there is a magnitude involved and
even though you can have more or less books, there's no direction here.
Therefore, it is not a vector since vectors must involve a direction.
In fact, quantities that involve a magnitude but no direction are called
scalars. Here's a fact that might come in handy:
every vector has a scalar associated with it and this makes sense because a
scalar quantity is one with only magnitude. But a vector involves a
magnitude as well as a direction so we can actually think of a vector as a
scalar quantity that also involves a direction. In fact, there is even a
special way to indicate the magnitude of a vector and that is ... for any vector V we
denote its magnitude by putting two lines on either side of it, like so. And
like we mentioned earlier, this arrow denotes that this is a vector. Good! Let's
go over one last example. We go online to check the weather and find that the wind
is blowing south at 30 kilometers an hour. Would this be
considered a vector quantity? Of course! It is and why? well that's because it
contains a magnitude which corresponds to the speed of the wind the wind could
be faster or slower. And it also tells us where the wind was blowing. In other
words, the direction of the wind. Again, magnitude and direction together give us
a vector. Awesome! Now we know what a vector quantity is. It's simply any
mathematical quantity that involves both a magnitude and a direction. We also
briefly went over what a scalar quantity is and how it is a quantity that only
represents magnitude. In the next lesson, we'll learn how to mathematically
represent vector quantities so until next time have a good one