VECTORS.

The word vector comes from a latin word which means 'one who carries, one who takes, one who transports'. As you can see, we can translate it as the conductor, bearer,…

A vector is a piece, part, portion or segment of a line which indicates more than just its length or magnitude; it also indicates direction and sense. 
From line  X (this would be direction), we take a piece, a part or segment and we mark its length with letters A and B. Then, we draw an arrow to the right to indicate its sense.

Vectors are represented with letters to determine its magnitude (initial point and terminal point). In the previous example, the initial point of the vector is found in A and the terminal point is B. Vectors are represented with letters (first the initial point, then the terminal point) and an arrow above them to represent its sense:

 

 Vectors can be written in just one uppercase or lowercase letter:  

Many people consider the words direction and sense to be the same. However, they are different. Direction is the line in which movement is created while sense indicates where this movement is performed (where we are going), to one side or the other.

Imagine a highway going from Irun to Madrid (and vice-versa, of course). In this case, we are referring to direction. Usually, a single highway can take you to different places. In other words, there can be more than two cities found in a highway. Nevertheless, there are two senses whenever we are traveling from one city to another: we are either going to Madrid or we are going to Irun. See the image below:

The train tracks (railroad) indicates direction. The train is either going on the sense of Madrid or the sense of Irún. There can be many directions (places where train tracks have been laid out) but there can only be two senses, no more.

8.11 In this exercise, you have some vectors you need to group:

1º  those which have the same magnitude
2º  those which have the same direction
3º  those which have the same sense

Answers: same magnitude: 

same direction: 

same sense: 

EXERCISES WITH VECTORS.

If vectors can be represented in graphics, we can also represent their sum graphically.

Adding vectors:
To add two or more vectors in a graph, we join their initial points just like you have it in F 1. As you can see, both coincide in point (0,0). Let's add these vectors:

The first vector has its origin (initial point) in (0,0) and its terminal point is in (2,2). The second vector  also has its origin (initial point) in (0,0) and its terminal point is in (5,2).

It is advisable for you to draw them in a coordinate axis. The sum can be performed in two ways:

fig.2

a) graphically

b) adding their components

Once you have drawn the vectors in their corresponding locations, trace a line parallel to starting from A and another line parallel to starting from B, as seen in F 2:

These two lines cut in point C. Join this point with the origin of both lines (initial point for both vectors) to form vector . Its initial point is located at (0,0) and its terminal point is in (7,4). This is the result of adding  

b) The terminal point of vector  is located at (2,2) and the terminal point of vector  is (5,2). The two values which define a point, in other words, the values of x and y are called components. If we add the components of x and y in an orderly fashion, we get (2+5,2+2) = (7,4). This is equal to the value we got previously in a graph.

8.12 Add vectors  and  graphically. Their lengths, directions and senses are shown in F 3 starting from point (1,2).

Solution:

In F 3, we have located the two vectors we want to add at a point different from (0,0); they are located at point (1,2).

We will solve this graphically by tracing parallel lines for both vectors starting from A and B, as we did in the previous case. They cut at point C. We join this point with O to get the resulting vector from adding both vectors:

 , its coordinates correspond to point (8,6).

However, this answer is not correct because we have started from point (1,2). This means we need to subtract the components of points (8,6) and (1,2); (8,6) - ( 1,2) = (8 – 1, 6 – 2) = (7,4). You have these values represented in F 4:

 

In the Y-axis, we see that point C has a value of 6. However, since it started from a value of 2 and not from (0,0), its real value is equal to 6 – 2 = 4. The same thing happens with the X-axis, which has a value of 8. Nevertheless, since it started from 1 and not from (0,0), its real value is equal to 8 – 1 = 7. The total resulting value is (7,4). The result is the same as the one obtained after adding its components.

Another way of calculating a sum of vectors graphically is to relocate the second vector at the terminal point of the first vector and joining the initial point of the first vector with the terminal point of the second. You will understand this better in F 5:

Here, we have placed vector  after vector .

Fig. 5

We join the terminal point of vector  with the initial point of vector  to get vector  . It corresponds to the sum of the components of both vectors:    :  (7,4).

8.13 In the first quadrant of a coordinate axis, draw two vectors and calculate their sum. Solve this using at least two different ways.

Solution:

In F 6, we traced two vectors , both with an initial point at (2,3) and terminal points at (3,5) and (5,2), respectively. In this case, we are representing vectors with a single uppercase letter.

The result is represented by vector = (6,4)

Adding the components, we get: (3+5, 5+2) = (8,7). However, this value does not originate from point (0,0) but from point (2,3). We need to subtract components (2,3) from components (8,7):

 (8 – 2, 7 – 3) = (6,4).

 

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