1

A vector has initial and final endpoints and respectively. Find the coordinates of

Solution

We know that the coordinates of a vector are obtained by subtracting the initial point from the final point

2

A vector has final and initial endpoints and respectively. Find the coordinates of

Solution

We know that the coordinates of a vector are obtained by subtracting the initial point from the final point

3

A vector has components . Find the coordinates of if endpoint is known

Solution

1 Since we don't know the coordinates of , we denote them by

.

2 We know that the coordinates of a vector are obtained by subtracting the initial point from the final point

3 We obtain two equations

4 We solve both equations and obtain that the coordinates of are

4

A vector has components . Find the coordinates of if endpoint is known

Solution

1 Since we don't know the coordinates of , we denote them by .

2 We know that the coordinates of a vector are obtained by subtracting the initial point from the final point

3 We obtain two equations


4 We solve both equations and obtain that the coordinates of are

5

Given the vector and two vectors equivalent to and , determine and knowing that and

Solution

1 Since are equivalent, then .

2 Since we don't know the coordinates of , we denote them by

.

3 We know that the coordinates of a vector are obtained by subtracting the initial point from the final point

4 We obtain two equations

5 We solve both equations and obtain that the coordinates of are

6 Solving in the same way as for , we have that .

6

Calculate the distance between points and

Solution

1 The formula for the distance between two points is

2 We substitute the values of and in the distance formula between two points and obtain

7

Calculate the distance between points and

Solution

1 The formula for the distance between two points is


2 We substitute the values of and in the distance formula between two points and obtain

8

Find the value of so that the distance between points and is 7

Solution

1 The formula for the distance between two points is


2 We substitute the values of and in the distance formula between two points and obtain


3 Squaring both sides and solving, we obtain

9

Find the value of so that the distance between points and is 8

Solution

1 The formula for the distance between two points is



2 We substitute the values of and in the distance formula between two points and obtain


3 Squaring both sides and solving, we obtain

10

If is a vector with components , find a unit vector with the same direction and sense

Solution

1 The formula for a unit vector is

2 We calculate the magnitude of

3 We substitute into the formula to obtain a unit vector

11

If is a vector with components , find a unit vector with the same direction and opposite sense

Solution

1 The formula for a unit vector is


2 We calculate the magnitude of


3 We substitute into the formula to obtain a unit vector


4 We are asked for the unit vector to have opposite sense, that is

 

12

Find a unit vector with the same direction as the vector

Solution

1 The formula for a unit vector is

2 We calculate the magnitude of

3We substitute into the formula to obtain a unit vector

13

Calculate the coordinates of so that the quadrilateral with vertices and is a parallelogram.

Solution

Ejercicio de vertice de un paralelogramo

1 The opposite sides of a parallelogram are equal in magnitude and direction, so we have

2 Since we don't know the coordinates of , we denote them by

.

3 We substitute the values of the vertices of the parallelogram into the vector equality

4 We obtain two equations

5 Solving the equations we obtain the sought coordinates

14

Find the coordinates of the midpoint of segment , with endpoints and .

Solution

1 The formulas for the coordinates of the midpoint are

2 We substitute the values of and into the two previous formulas

3 The midpoint is .

15

Find the coordinates of point , knowing that is the midpoint of , where .

Solution

1 The formulas for the coordinates of the midpoint are

2 We substitute the values of and into the two previous formulas and calculate the first coordinate of

3 The second coordinate of is

4 Finally

16

Determine whether points and are collinear.

Solution

1 Points are collinear if the slopes of segments and are equal.

2 Since both slopes are equal, then the three points are collinear.

17

Determine whether points and are collinear.

Solution

1 Points are collinear if the slopes of segments and are equal.

2 Since both slopes are not equal, then the three points are not collinear.

18

Calculate the value of so that points are collinear.

Solution

1 Points are collinear if the slopes of segments and are equal.

 

 

2 Since both slopes are equal, we equate both expressions and solve for

19

Given points and , find a point collinear with and , such that

Solution

1 We start from the given condition and obtain an equality

 

 

2 We equate both expressions coordinate by coordinate and obtain

 

 

3 We solve both equations to obtain the coordinates of

 

 

20

Given a triangle with vertices and , find the coordinates of the centroid

Solution

1 The formula to find the centroid is

 

2 Substituting the values of the vertices of the triangle we obtain

 

21

Given a triangle with two of its vertices and the centroid , calculate the third vertex

Solution

1 The formula to find the centroid is

 

 

2 Substituting the values of the centroid and the vertices of the triangle we obtain two equations

 

 

3 We solve both equations and obtain the third vertex .

22

Find the symmetric point of with respect to

Solution

1 We denote by the symmetric point of , then it holds that

 

2 Substituting the values of the points, we obtain two equations corresponding to the coordinates of the vectors

 

 

3 We solve both equations and obtain .

 

23

Find the symmetric point of with respect to

Solution

1 We denote by the symmetric point of , then it holds that

 

2 Substituting the values of the points, we obtain two equations corresponding to the coordinates of the vectors

 

 

3 We solve both equations and obtain .

24

What points and divide the segment with endpoints and into three equal parts?

Solution

Ejercicio de triseccion de un segmento

1 In vector notation we have

2 Substituting the values of the points, we obtain two equations corresponding to the coordinates of the vectors

3 We solve both equations and obtain .

4 To find the coordinates of we use the condition

5 Substituting the values of the points, we obtain two equations corresponding to the coordinates of the vectors

6 We solve both equations and obtain .

25

If segment with endpoints is divided into four equal parts, what are the coordinates of the division points?

Solution

Ejercicio de dividir un segmento en cuatro partes iguales

1 We note that is the midpoint of segment

2 is the midpoint of segment

3 is the midpoint of segment

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.