SLC S23 Week5 || Coordinates&Vectors

Hello everyone! I hope you will be good. Today I am here to participate in the contest of @sergeyk about Quadrilaterals. It is really an interesting and knowledgeable contest. There is a lot to explore. If you want to join then:

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Join Here: SLC S23 Week5 || Coordinates&Vectors

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Designed with ^Canva

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

The number line is the simple mathematical concept. According to this concept the numbers are placed at the equal distance on the straight line. It helps in the addition, subtraction and other mathematical operations. The number line has an origin at 0 and on the right side there are positive numbers and on the left side there are negative numbers.

I still remember that in the school I first used number line for the purpose to place positive and negative numbers and then I learned addition of the number on the number line and similarly subtraction. I am not sure about the exact class but I can say that it was in the primary classes.

The coordinate plane is the extension of the number line in the two dimensions. It uses two perpendicular lines. The line which is horizontal that is known as the x-axis and the line that is vertical that is known as the y-axis. The coordinate plane is used to locate the points in the order pair form. In the order pair form first the x coordinate is written and then the y coordinate is written such as (x,y).

I think I learned about the coordinates in the 6th class when I started reading head to tail rule of the vectors along with the vector addition and vector difference. The coordinate plance has four quadrants as I have shown in the picture. Each quadrant has coordinates in this order:

• 1st Quadrant (x, y)
• 2nd Quadrant (-x, y)
• 3rd Quadrant (-x, -y)
• 4th Quadrant (x, -y)

So any point which is placed in any quadrant follow this order for the values of the coordinates.

If I talk about the difficulty yes I will say that as a beginner it was difficult for me and it was not easy for me to tackle with a new thing but with the careful teaching method of my teacher and attention from me made me able to understand its concepts.

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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))

To connect the pairs of points to form the vectors first we need to place all the given points on the plane and then I will construct the vectors on the coordinate plane by connecting the pair of the vectors. I will try to construct unique vectors which can be in the different locations in the plane.

Here I have placed all the points on the plane. Now I will connect them to create some vectors on the plane. The formula to form the vectors from the given points is AB = B - A = (x_B - x_A, y_B- y_A). So my each vector should

I will build these vectors by joining the pairs of the points:

• AB = B - A = (4−(−3),7−11)=(7,−4)

• AC = 𝐶−𝐴 = (0−(−3),4−11)=(3,−7)

• BD = D - B = (4−4,0−7)=(0,−7)

• CF = F - C = (11−0,3−4)=(11,−1)

• DE = E - D = (−4−4,−7−0)=(−8,−7)

• CE = E - C = (−4−0,−7−4)=(−4,−11)

So I have developed the 6 vectors by joining different points and all the vectors are accurate with respect to the formula. After drawing all the vectors a bow and arrow has formed 😂 🤣. I was not expecting that while drawing the vectors I will achieve this. But it is looking beautiful.

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

Let us start placing these three vectors on the plane and then I will construct the sum of vector of all these three vectors.

Here you can see the three vectors on the plane. Two vectors a and c both are in the first quadrant and both the coordinates are positive. The vector b is in the third quadrant.

I have drew the vector a and now I will add vector b in it. The coordinates of the vector a are (3,7) while adding vector b with coordinates the result is (3-1, 7-3) = (2, 4). So I will place vector from the head of the vector a up to (2 , 4).

Here you can see I have drawn the vector b by adding it into the vector a with the help of the head to tale rule. Now from the coordinates (2 , 4) I will place the vector c(1, 5). If we add these coordinates we get (3 , 9) coordinates so our next vector should start from the head of the vector b and it should go to towards the coordinates ( 3 , 9 ).

Here you can see I have added the 3rd vector c as well. This vector is ending at the point D with coordinates (3 , 9). Now to find the resultant vector of all these three vectors I will join the tail of the vector a with the head of the vector c.

Finally I have drawn the resultant vector or the sum vector a+b+c, it is reprsented by the blue dots and dash arrow starting from the origin and ending at the point D. I have added all these vectors with head to tail rule.

Mathematical Verification

Here is the mathematical verification of the addition of a+b+c.

a+b+c = (3 - 1 + 1 , 7 - 3 + 5) = (3 , 9)

The resultant vector has the coordinates (3 , 9) which is same as the mathematical result.

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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.

I have placed two random points on the plane. One point is in the first quadrant and the second point is in the second quadrant.

Here You can I have drawn segments from the points ad in this way I have found out the coordinates of the points. The point p has (2 , 5) and the point Q has (-3 , 1) coordinates.

The vector PQ is created by subtracting the coordinates of P from Q.

PQ = (Q_x - P_x, Q_y - P_y)
PQ = (-3-2, 1 — 5) = (−5, −4)

Now in order to find the vector which is twice the constrcuted vector PQ I will simply multiply the coordinates of the vector PQ by 2 and then I will draw the new vector.

So by multiplying the vector PQ with 2 we get these new coordinates 2PQ = 2×(−5,−4) = (−10,−8).

I have found out the 2PQ vector starting from the point P as well as starting from the origin. In both the cases the value of the twice vector remains same with the slight difference in the drawing. In the left side the double vector overwrites the previous vector and it is extended upto point E. In the right the double sized vector is starting from the origin of the plane.

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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.

I have taken three vectors as given below:

• a = (3, 4)
• b = (-2, 5)
• k = (1, -3)

I will plot all these vectors on the plane and then I will perform the relevant tasks. The origin of all these vectors will be at (0 , 0).

Here you can see all the three arbitrary vectors. I have drawn all the vectors as I already determined their arbitrary positions.

Now I will find the u = a + b. By using vector addition u = a+b = (3,4) + (−2,5) = (3−2,4+5) = (1,9).

Here you can see I have found u = a + b. To add vector b to vector a I drew the vector b starting from the head of the vector a and it ended at the point F with the coordinates (1 , 9). Then I joint the tail of the vector a with the head of the vector b and this resultant vector u is formed and it is highlighted with the red dotted vector.

Now I will find the v = b - k. By using vector subtraction v = b−k = (−2,5) − (1,−3) = (−2−1,5−(−3)) = (−3,8).

Here you can see I have found v = b - k. I used head to tail rule. I drew the vector k starting from the head of the b and as I am subtracting the direction of the vector has changed from the coordinates (1 , -3) to (-1, 3) this vector ended at the point G. Then I connected the Origin point to the point G and I got the vector v which has the coordinates (-3, 8).

I have hidden all the other vectors and now we have only two vectors u and v on the plane. Now I will construct u + v. By adding the coordinates we get:

u+v=(1,9)+(−3,8)=(1−3,9+8)=(−2,17)

I added the vector v to u using the head to tail rule so I started the vector v from the head of the vector u and it ended at the point H. Then I connected the tail of the vector u to the head of the vector v and got the vector u+v with the coordinates (-2 , 17) which is correct as we found mathematically.

Now I will construct u - v. By subtracting the coordinates we get:

u−v=(1,9)−(−3,8)=(1+3,9−8)=(4,1)

I have subtracted the vector v from vector u using head to tail rule on the plane. The direction of the vector v was changed because of the subtraction. I drew it from the head of the vector u and it ended at the point I. I connected the tail of the vector u to the head of the vector v with point I and it returned me u-v vector with the coordinates (4, 1). The vector u-v is represented by the green dashes arrow. These coordinates are equal to the mathematically calculated coordinates.