Hello.
Welcome to the mini-course on vectors.
This course is meant for students who plan to win the Nobel prize.
It gives you all the basic knowledge you need to apply vector properties in Mechanics!
If you don't plan to win the Nobel prize, but you want to pass your exam in mechanics,
this course is also exactly what you need.
Take a look at this airplane!
It is surrounded by various arrows.
These arrows have a starting point, a direction and a length.
This picture of a bridge also shows such arrows.
These arrows are not drawn just to give the picture a scientific flavour.
They are very useful for making relevant computations.
In this video, we introduce the basic concepts of these arrows, called vectors in mathematics.
Vector is a latin word, which means carrier.
It was first used by 18th century astronomers investigating planet rotation around the Sun.
A vector is often used in geometry and physics to represent physical quantities
that have both magnitude and direction.
Think of physical quantities like speed, movement, acceleration and force!
Consider this pool table.
If you want to indicate a location on the pool table, this can be done by fixing an origin,
and a horizontal and vertical position of the location, relative to the origin O.
For example: the point (6,4) and the point (-8,2).
If we want to describe a movement on the pool table, this can be done using a vector.
The movement from (6,4) to (-8,2) can be represented by an arrow starting at (6,4) and ending at (-8,2).
If we want to...
If we denote the point (6,4) by A and (-8,2) by B,
the vector in this picture is denoted as AB with a little arrow on top.
There are various operations that can be applied to vectors.
In this video, we consider two of these: scalar multiplication and addition.
We start with addition.
These operations can be viewed from a geometric point of view as well as from an algebraic point of view.
We start with the geometric view.
So consider three points in the plane: A, B and C.
Construct the vectors AB and BC .
The sum of these is then obtained by following the two arrows, first AB and then BC.
This gives you AC .
Now, what to do if the two vectors are not nicely attached,
in the sense that head and tail of the two vectors are not connected?
Well, then it is important to realize that for vectors, only length and direction are important.
In that sense, the two vectors given here are actually equivalent.
When we look at a vector u or v, we have to stress that we mean vectors, rather than numbers.
Unfortunately, there is not just one universally accepted notation to do this.
Often, the u is underlined to stress that it is a vector and not a number.
Other people use u with a little arrow on top and other people use boldface.
In this video, we use the boldface notation.
Remember this is just a convention and don't panic when you see people using other ways
to indicate the difference between a scalar and a vector.
Ok. Now, since two vectors are considered equal whenever their length and direction coincide,
adding two vectors can be done by shifting the second vector such that its tail starts
at the head of the first vector.
In this picture, first shift the vector v such that its starting point is exactly the endpoint of u
and then add the vectors as before .
In practice however, if you want to add two vectors, usually you first shift their starting points to the origin.
Then the sum of the two vectors can be obtained by using the so called parallelogram law.
Vectors can also be multiplied by a number, often called scalar in this context.
Briefly, multiplying a vector by 3, means its direction is left unchanged
and its length is multiplied by this number 3.
For instance, consider this vector v.
3 times v is given by this.
Multiplying v by minus 3, on the other hand, will give a vector which has opposite direction to v,
and its length is still 3 times that of v.
Ok, let us now look at vector addition and scalar multiplication of vectors
from an algebraic point of view.
The standard algebraic representation of a vector in the two dimensional space,
is by its end point, assuming the vector starts at the origin.
The coordinates of the end point are stacked on top of each other
and collected between two tall brackets.
For instance (4,1) and (-3,5).
Using this representation, addition of two vectors is quite natural.
The numbers 4 and 1 are called components of the vector (4,1).
Geometrically, we see that u plus v starts at the origin and ends in (1,6) .
The representation of this sum vector is (1,6).
Note that the first component 1 of the sum vector u+v is simply the sum of the first components
4 and minus 3 of the vectors u and v.
Similarly, the second component 6 of the sum vector u+v is just the sum of the second components
1 and 5 of the vectors u and v.
Multiplying the vector u equals (4,1) by 2 gives (8,2), of course.
Multiplying u by minus a half, leads to (-2,-1/2).
This can be seen from the geometrical representation as well as from the algebraic representation.
The examples up till now were given in two dimensions.
In three dimensions, you can take exactly the same approach.
A vector has a direction and a length.
It can be represented by an arrow.
Here you see the vector u with end point (1, 2, 3).
Adding the vector (1, 2, 1), leads to the vector (2, 4, 4).
This can be seen by moving the second vector's tail to the first vector's head
and travelling along the connected arrows.
Alternatively, the sum of u and v can easily be obtained by adding just the components of the vectors.
In this video we defined vectors in two and three dimensional space
both geometrically by magnitude and direction, and algebraically by components.
You also learned how to apply the following operations:
addition of two vectors and multiplication of a vector with a scalar.
In the next video you will learn more about the length of a vector and distances between points.
Stay tuned!
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