Professor Dave here, let's talk about
scalars and vectors.
Physics is all about some
pretty amazing stuff, from subatomic
particles to everything you see in space.
But before we dive into all that, just as
with any other science, we have to get a
few things out of the way in terms of
language and convention. To learn about
science we have to know how people do
science, and how science is communicated,
so we must first know about things like
the units of measurement that the
scientific community uses, scientific
notation which is used to report very
small and very large numbers that will
be common in physics, and dimensional
analysis for converting between units.
Luckily we've already gone over these in
my chemistry series, but in case you
missed those tutorials you may want to
check them out at some point. Beyond
these basics, one other prerequisite for
studying physics is the understanding of
scalars and vectors. To put it briefly, a
scalar is a quantity that communicates a
particular size or magnitude. These can
be things like the mass of a book, the
amount of time you spend watching this
clip, or the temperature outside today.
These are numerical quantities along
with any relevant units that simply
answer the question: How much? A vector is
different because a vector has not only
magnitude but also direction, so it
answers both: How much? And: Which way?
Vectors will be very important in
physics because when we look at forces
we want to know how much force is being
applied but also in which direction.
These forces will always be represented
by vectors where the direction the arrow
is pointing is important information
just as much as the length of the arrow
which will indicate the magnitude of the
vector. To denote vectors we usually use
letters in bold font with an arrow on
top. As we learn more about physics we
will use vectors to predict the motion
of objects, and if there are several
forces acting on
an object, as there often are, we will
need to be able to do certain operations
with vectors in order to perform our
calculations, so let's learn how to do
simple vector operations. If we want to
add two vectors together it's pretty
simple, we just make the second vector
start where the first vector ends and
then create a new resultant vector that
goes from the start of the first to the
end of the second. In lining up two
vectors we have to make sure they retain
their direction, as that is important
information. If the vectors point in the
same direction their sum will just be a
longer vector, and the magnitude of this
resultant vector will simply be the sum
of the magnitudes of the original
vectors, so in this case 4 plus 3 equals 7.
Sometimes there is an angle between
the two vectors, like these perpendicular
vectors. In this case, when we draw the
resultant vector it will be the
hypotenuse of a right triangle. The
magnitude of this resultant vector is
not the sum of the other two magnitudes,
this one can be found by using the
Pythagorean theorem, which we may
remember from algebra class. A squared
plus B squared equals C squared. Plug in
A and B, square them and find the sum, and
then we take the square root to get C,
which in this case will be 5. When we
are looking at right triangles we can
also use trigonometric functions to find
the unknown angles, all we need to do is
choose an angle which we represent with
the Greek letter theta, and label the
sides of the triangle as opposite, the
leg directly opposite the angle, adjacent,
the leg right next to the angle, and the
hypotenuse, which is always the longest
side. We can use the lengths of these
sides to find the sine, cosine, and
tangent of this angle. We will cover
these functions in more depth in the
upcoming mathematics course, but for now
all we need to remember is the mnemonic
device: SOHCAHTOA. Sine theta is equal to
opposite over hypotenuse,
in this case that would be 3/5.
Cosine theta is equal to adjacent
over hypotenuse, which here will be 4/5,
and tangent theta is opposite
over adjacent, or 3/4. To solve
for theta we can use any of these
equations. Using this one we can take the
inverse sine of both sides, that makes
theta equal to the inverse sine of
3/5. Just put 3/5 or 0.6 into
your calculator and press the inverse
sine button which may look like any of
these expressions, and we get about 36.9
degrees. Trigonometric functions like
these relate the angles of a right
triangle to the length of the sides and
we will use them a lot in physics, but it
usually won't be more complicated than
this so don't worry too much about it.
Also, if you are using a graphing
calculator take a moment to familiarize
yourself with the different settings it
uses to report angles, which are degrees
and radians. If you are more familiar
with degrees make sure that setting is
selected and we will learn about radians
another day.
Now that we understand vector addition
let's quickly go through a few other
operations. Vector subtraction will be
similar to vector addition
except that when subtracting vectors we
will again line them up head to tail but
we will invert the direction of the
second vector. This is the same as
multiplying the vector by a scalar of
negative 1.
This way the magnitude of the vector
remains the same but its direction is
reversed, so instead of A plus B which
looks like this, A minus B would look
like this, which is essentially A plus
negative B. We find the resultant vector
and that's all there is to it.
Another thing we might want to do is
break up a vector into X and Y
components.
This is useful when looking at
projectile motion. If a ball is thrown at
an angle, its velocity can be represented
with a vector, but because gravity is
operating on the ball in the Y-direction
pulling it down to the earth, we will
want to be able to
separate its vertical motion from its
horizontal motion, so we can take this
vector and split it up into components.
All we need to know is the magnitude of
the vector and its angle from the
horizontal, then we can use trigonometry
to find the length of the legs of a
hypothetical right triangle. Sine of
theta gives us opposite over hypotenuse
and cosine theta gives us adjacent over
hypotenuse. On both of these we can
evaluate the trig function and multiply
both sides by the hypotenuse, which is
the value we know, to get the legs of the
triangle which will be the X and Y
components of the vector. We will see
this in more detail later.
Vectors can also be multiplied by
scalars or other vectors. When
multiplying a vector by a scalar we will
simply multiply the magnitude of that
vector by the scalar while keeping the
direction the same, so if multiplying
this vector by 2, its length will double.
Two vectors can also be multiplied but
that will be more complicated, so we will
go over that another time.
Let's try one example to make sure we
can do these kinds of calculations. Say
you are in a boat traveling across a
river and the boat typically moves at 4
meters per second in still water. But the
current is pushing you downstream at a
rate of one meter per second.
What is the speed and direction of the
boat? To answer this and almost any other
problem we study in physics it is best
to draw a diagram first. Here's a
top-down view of the river and the boat.
Let's draw a vector that is horizontal
with a magnitude of 4. We then need
another vector to represent the current.
That will point straight down and have a
magnitude of 1. To find out the speed
and direction of the boat we have to add
these vectors together. In order to do
this we can move the vector that
represents the current over here so that
we are adding head to tail and then find
the resultant vector, or we could have
left it where it was, made a rectangle
using the two vectors and then just drawn
the diagonal.
Either way this vector sum will have a
magnitude of root 17. To find the angle
from the horizontal we can use any trig
function we like. Say we use tangent.
tangent theta will be opposite over
adjacent, or one-fourth. Taking the
inverse tangent, theta must be 14 degrees,
so the boat is moving at a speed of
root 17 meters per second in the
direction of 14 degrees off the
horizontal. It may seem abstract right
now but we will be using this sort of
approach in order to understand motion
in almost any situation you can imagine
because it's just about using vectors to
represent the forces acting on any
object. Now we are ready to move on to
the physics, but first let's check comprehension.
Thanks for watching, guys.
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