Until now, we have been drawing in GeoGebra on a blank sheet without any lines. But for this topic, we need to get familiar with coordinates and the coordinate plane—or simply recall them for those who studied well in school.

Coordinate plane.

Right-click on the white sheet and enable the axes by selecting "Show Axes" and "Show Grid". After this, the coordinate axes will appear on the sheet: the horizontal OX and the vertical OY. These are coordinate lines that intersect at a right angle at point O.

The arrows indicate the positive direction. To the right of point O are positive numbers, while to the left are negative numbers. Similarly, the upward arrow means that numbers above O are positive, and those below are negative.

Coordinates.

On this coordinate plane, you can place points and mark their coordinates.

Let's place a point A somewhere on the plane and drop a perpendicular to the OX axis. The base of this perpendicular will be the number 2. Similarly, we drop a perpendicular to the OY axis, where the base will be 4.
In this case, we say that point A has the coordinates (2,4), and we write it as A(2,4).
Similarly, for point B, the coordinates will be (-3,-1).

Vectors

If we now go to the Lines section, select the Vector tool, and first click on point A and then on point B, we will construct the vector AB.
When naming a vector, the first point represents its starting point, and the second point represents its endpoint. Since a vector is a directed segment, it has a starting point and an endpoint.
A vector has its own coordinates, which are found by subtracting the coordinates of the starting point from the coordinates of the endpoint:
AB=(−3−2,−1−4)=(−5,−5)
So, the vector AB has the coordinates (-5, -5).
When writing a vector, it is essential to list the starting point first and the endpoint second. The vector AB is not the same as the vector BA.
A vector can also be represented using a single lowercase letter, such as u(-5, -5).

Let's place points C(0,5) and D(-5,0) on the coordinate plane and construct the vector CD.

The coordinates of vector CD are calculated as follows:
CD(-5-0, 0-5)=CD(-5,-5)
Thus, we have obtained another vector with the same coordinates as AB.

When two vectors have the same coordinates, they are called equal vectors. So, in this case, we say that AB = CD.

Another tool related to vectors in GeoGebra is the "Vector from Point" tool, which allows you to place a vector starting from a specific point.
To explore this, let's first construct a random vector AB. Then, we will place four arbitrary points C, D, E, and F on the coordinate plane.

Select the "Vector from Point" tool.
Click on the vector and then choose the point from which to place it.
Again, click on the vector and then choose the point from which to place it, and so on.
Then, we will move our initial vector — and all the other vectors will move as well. (In reality, there are no other vectors; it's the same vector, just applied to different points, attached to different points.)

Collinear vectors

Collinear vectors are those that lie on parallel lines or on the same line. Collinear vectors can be either similarly directed or oppositely directed.

For example, if the vector KL(4, -6) is given, the opposite vector LK will have coordinates LK(-4, 6).

Addition of Vectors

We will construct two vectors u=AB and v=CD.
How to construct their sum? There are two methods for this.

The triangle method.

We will place vector v starting from the end (point B) of vector u.

We will construct the vector AB' = u + v, this is the sum of the vectors.

The parallelogram method.

This method consists of a greater number of operations.

We will place vector v from the origin of vector u, forming the vector AA'.

Now, from the end A' of the formed vector AA', we will place vector u. This will form the vector A'A''.

We will construct the vector AA'' – this is the sum of the vectors u and v.

Subtraction of Vectors

To create a vector that is the difference of two vectors, we can also use two methods – the triangle method and the parallelogram method.

As in the case of addition, we will place two vectors from one point (we align the starting points of the vectors).

Now, we will form a vector by connecting the ends of the vectors, and it will be directed towards the vector being subtracted, that is, towards the vector from which we are subtracting.

If we align the starting points of the vectors at one point and complete a parallelogram on these vectors, one diagonal will be the sum of these vectors, and the other diagonal will be the difference of these vectors.

homework

Task 1. Tell about the number line and the coordinate plane. When did you first learn about coordinates? Was it difficult?

Task 2. Connect the pairs of points with vectors. Which vectors are formed? Show these vectors on the plane. A(-3, 11), B(4,7), C(0,4), D(4,0), E(-4,-7), F(11,3).
If all vectors are shown, there will be too many of them, so display only a few. For example 5 or 6))

Task 3. Place the vectors a(3,7), b(-1,-3), and c(1,5) on the plane.
Construct the vector a + b + c.

Task 4. Place two random points and determine their coordinates.
Create a vector from these points and write its coordinates.
Construct a vector that is twice as large as the created one.

Task 5. Construct three arbitrary vectors: a, b, k.
Build the vectors u = a + b and v = b - k.
Then construct u + v and u - v.

For all tasks: 1.8 points.
Additional point for aesthetic appearance, accuracy, and neatness of drawings, labels, and annotations: 1 point.
Bonus task – Find out what a vector projection is, explore it, and construct the projection of one vector onto another.

Rules for Participation

Title of your work: SLC S23 Week5 || Coordinates&Vectors

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Submission Period: From Monday (March 17/2025), to Sunday (March 23/2025).

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