This video is about vectors and scalers. We will start by discussing the difference between
vectors and scalers. Then we will see how to represent, add and subtract vectors, and we
will finish with talking about the resolution of vectors into perpendicular components.
Let's see what vectors and scalers are. A scalar is a quantity that has magnitude, but
no direction, and the vector is a quantity that has both magnitude and direction.
For example, if I say that I'm walking at 5 kilometers per hour, I am using a scalar
because I'm only providing a magnitude, 5 kilometers per hour. On the other hand, when I say
that I'm walking east at 5 kilometers per hour, I am using a vector because I'm using
magnitude, 5 kilometers per hour, and the direction east. You will come across many scalers
and vectors in physics. Let's see a few examples of each. Math, time, distance, temperature,
and speed are all scalers. In our example, 5 kilometers per hour is speed. While force, acceleration,
displacement, magnetic field strength, and velocity are all vectors. When I say that I'm walking
east at 5 kilometers per hour, I am talking about my velocity. Let's see how vectors are represented.
Vectors are represented by arrows like this, or maybe like this. We often write the name of the vector
next to it. For example, A, and to show that this is a vector and not a scalar, we can add
an arrow on top. In textbooks, vectors are often represented with letters written in bold.
We can also write the value of the quantity represented by the vector next to the vector.
So, for instance, if this is a force vector, representing three new tons, I would write
three new tons next to it. Note that the magnitude represented by a vector is proportional to its
length. So, if I want to draw a vector that represents a force of six new tons, it would
have to be twice as long as the three new tons vector. Next, let's discuss graphical vector
addition and subtraction. We are going to work with vector A and vector B. Let's see how to add
and subtract these vectors. There are many different methods for vector addition and subtraction.
Here, I will just show one of these. To add A and B, I will redraw the vectors, placing the
starting point of B at the end point of A. Based on the definition of vector addition, A plus
B is this vector. Let's see subtraction. Here, I will redraw the vectors again, but instead of
B, I will draw a negative B at the end point of A like this. So, the black vector is still
A, but the blue vector here is negative B, because it has the same magnitude as B, but points
in the opposite direction. So, using vector addition, this vector here is a plus negative B, which
is the same as A minus B. Finally, we will see hat result vectors into perpendicular components.
To do so, we will use this diagram and formulae. Let's see two examples. In the first one,
we will resolve A into a horizontal and a vertical component and find the magnitude of these components.
The horizontal component looks like this and the vertical component like this. Using the
formulae, below the diagram in the blue box, for AH, the horizontal component, we get
the magnitude of A, which is 5, times cosine of the angle, and this is 30 degrees. This gives us
approximately 4.33 centimeters. For AV, the vertical component, we get 5 times sine 30, which is equal
to 2.5 centimeters. In example, 2, and object rests on a slope, and we have to find the magnitude
of the component of the objects weight that acts perpendicular to the slope. Let's start
by drawing this situation. The slope looks like this, and here is the object resting on the slope.
The weight of the object is pointing downwards. Let's resolve this weight into two components,
one parallel and one perpendicular to the slope. The parallel component looks like this, and
the perpendicular component looks like this. Due to the symmetry in the diagram, this angle
here is also 20 degrees. I won't go into the mathematical details of why this is so here.
Let's use the diagram in the blue box on the right. Imagine rotating the three vectors from the
diagram that we have drawn until vector W lines up with vector A. Doing this, you should see that
our purple vector lines up with AV, and our red vector lines up with age. In other words,
the red vector corresponds to age. The red vector is the perpendicular component of the weight,
and now we can calculate this magnitude, using the formula for age. So for the perpendicular
component, we get the magnitude of the weight, and since the mass of the object is 10 kilograms,
its weight will be 10 times the gravitational acceleration, so 10 times 9.81, which is 98.1
Newton's. If this is not fully clear at the moment, don't worry. For now, just focus on understanding
the work that we do with the vectors. So for the perpendicular component, we get 98.1 times
cosine 20 degrees, and this is approximately 92.2 Newton's. Let's summarize what we have learned.
To start, we looked at the definition of scalars and vectors. We mentioned that you will come across many
scalars and vectors in physics, and we listed a few examples for scalars and vectors.
We saw that vectors are represented by arrows, and that the magnitude of the quantity that
the vector represents is proportional to the length of the vector, and we discussed that to add
and subtract vectors graphically, we should place one vector at the endpoint of the other.
Finally, we saw two examples of resolving vectors into perpendicular components, where
we used this formula. And we have concluded our learning about vectors and scalars.
With this video, we have wrapped up subtopic 1.3. In the first video of subtopic 2.1 motion,
we will discuss distance and displacement.